International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 248-258
Section 4.2.6.3. Modern experimental techniques
D. C. Creaghb
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The atomic scattering factor enters directly into expressions for such macroscopic material properties as the refractive index, n, and the linear attenuation coefficient, . The refractive index depends on the dielectric susceptibility χ through
where
and
is the number density of atoms of type j.
The imaginary part of the dispersion correction for the case where
is related to the atomic scattering cross section through equation (4.2.6.23)
.
Experimental techniques that measure refractive indices or X-ray attenuation coefficients to determine the dispersion corrections involve measurements for which the scattering vector, , is zero or close to it. Data from these experiments may be compared directly with data sets such as Cromer & Liberman (1970
, 1981
).
Other techniques measure the intensities of Bragg reflections from crystalline materials or the variation of intensities within one particular Laue reflection (Pendellösung). For these cases, , the reciprocal-lattice vector for the reflection or reflections measured. These techniques can be compared only indirectly with existing relativistic tabulations, since these have been developed for the
case. Data are available for elements having atomic numbers less than 20 in the non-relativistic case (Wagenfeld, 1975
).
The following sections will discuss some modern techniques for the measurement of dispersion corrections, and an intercomparison will be made between experimental data and theoretical calculations for a representative selection of atoms and at two extremes of photon energies: near to and remote from an absorption edge of those atoms.
X-ray interferometer techniques
are now used extensively for the measurement of the refractive index of materials and hence . All the interferometers are transmission-geometry LLL devices (Bonse & Hart, 1965b
, 1966a
,b
,c
,d
, 1970
), and initially they were used to measure the X-ray refractive indices of such materials as the alkali halides, beryllium and silicon using the characteristic radiation emitted by sealed X-ray tubes. Measurements were made for such characteristic emissions as Ag
, Mo
, Cu
and Cr
by a variety of authors (Creagh & Hart, 1970
; Creagh, 1970
; Bonse & Hellkötter, 1969
; Bonse & Materlik, 1972
).
The ready availability of synchrotron-radiation sources led to the adaptation of the simple LLL interferometers to use this new radiation source. Bonse & Materlik (1975) reported measurements at DESY, Hamburg, made with a temporary adaptation of a diffraction-beam line. Recent advances in X-ray interferometry have led to the establishment of a permanent interferometer station at DESY (Bonse, Hartmann-Lotsch & Lotsch, 1983b
). This, and many of the earlier interferometers invented by Bonse, makes its phase measurements by the rotation of a phase-shifting plate in the beams emanating from the first wafer of the interferometer.
In contrast, the LLL interferometer designed by Hart (1968) uses the movement of the position of lattice planes in the third wafer of the interferometer relative to the standing-wave field formed by the recombination of two of the diffracted beams within the interferometer. Measurements made with and without the specimen in position enabled both the refractive index and the linear attenuation coefficient to be determined. The use of energy-dispersive detection meant that these parameters could be determined for harmonics of the fundamental frequency to which the interferometer was tuned (Cusatis & Hart, 1975
, 1977
). Subsequently, measurements have been made by Siddons & Hart (1983
) and Hart & Siddons (1981
) for zirconium, niobium, nickel, and molybdenum. Hart (1985
) planned to provide detailed dispersion curves for a large number of elements capable of being rolled into thin foils.
Both types of interferometers have yielded data of high quality, and accuracies better than 0.2 electrons have been claimed for measurements of in the neighbourhood of the K- and L-absorption edges of a number of elements. The energy window has been claimed to be as low as 0.3 eV in width. However, on the basis of the measured values, it would seem that the width of the energy window is more likely to be about 2 eV for a primary wavelength of 5 keV.
Apparently, the ångström-ruler design is the better of the two interferometer types, since the interferometer to be mounted at the EU storage ring is to be of this type (Buras & Tazzari, 1985).
Interferometers of this type have the advantage of enabling direct measurements of both refractive index and linear attenuation coefficients to be made. The determination of the energy scale and the assessment of the energy bandpass of such a system are two factors that may influence the accuracy of this type of interferometer.
One of the oldest techniques for determining refractive indices derives from measurement of the deviation produced when a prism of the material under investigation is placed in the photon beam. Recently, a number of groups have used this technique to determine the X-ray refractive index, and hence .
Deutsch & Hart (1984a,b
) have designed a novel double-crystal transmission spectrometer for which they were able to detect to high accuracy the angular rotation of one element with respect to the other by reference to the Pendellösung maxima that are observed in the wave field of the primary wafer. In this second paper, data gained for beryllium and lithium fluoride wedges are discussed.
Several Japanese groups have used more conventional monochromator systems having Bragg-reflecting optics to determine the refractive indices of a number of materials. Hosoya, Kawamure, Hunter & Hakano (1978; cited by Bonse & Hartmann-Lotsch, 1984) made determinations of
in the region of the K-absorption edge for copper. More recently, Ishida & Katoh (1982
) have described the use of a multiple-reflection diffractometer for the determination of X-ray refractive indices. Later, Katoh et al. (1985a
,b
) described its use for the measurement of
for lithium fluoride and potassium chloride at a wavelength near that of Mo
and for germanium in the neighbourhood of its K-absorption edge.
Measurements of the linear attenuation coefficient over an extended energy range can be used as a basis for the determination of the real part of the dispersion correction
because of the Kramers–Kronig relation, which links
and
. However, as Creagh (1980
) has pointed out, even if the integration can be performed accurately [implying the knowledge of
over several decades of photon energies and the exact energy at which the absorption edge occurs], there will still be some ambiguity in the result because there still has to be the inclusion of the appropriate relativistic correction term.
The experimental procedures that must be adopted to ensure that the linear attenuation coefficients are measured correctly have been given in Subsection 4.2.3.2. One other problem that must be addressed is the accuracy to which the photon energy can be measured. Accuracy in the energy scale becomes paramount in the neighbourhood of an absorption edge where large variations in
occur for very small changes in photon energy
.
Despite these difficulties, Creagh (1977, 1978
, 1982
) has used the technique to determine
and
for several alkali halides and Gerward, Thuesen, Stibius-Jensen & Alstrup (1979
) used the technique to measure these dispersion corrections for germanium. More recently, the technique has been used by Dreier et al. (1984
) to determine
and
for a number of transition metals and rare-earth atoms. The experimental configuration used by them was a conventional XAFS system. Similar techniques have been used by Fuoss & Bienenstock (1981
) to study a variety of amorphous materials in the region of an absorption edge.
Henke et al. (1982) used the Kramers–Kronig relation to compute the real part of the dispersion correction for most of the atoms in the Periodic Table, given their measured scattering cross sections. This data set was computed specifically for the soft X-ray region
.
Linear attenuation coefficient measurements yield directly and
indirectly through use of the Kramers–Kronig integral. Data from these experiments do not have the reliability of those from refractive-index measurements because of the uncertainty in knowing the correct value for the relativistic correction term.
None of the previous techniques is useful for small photon energies. These photons would experience considerable attenuation in traversing both the specimen and the experimental apparatus. For small photon energies or large atomic numbers, reflection techniques are used, the most commonly used technique being that of total external reflection. As Henke et al. (1982) have shown, when reflection occurs at a smooth (vacuum–material) interface, the refractive index of the reflecting material can be written as a single complex constant, and measurement of the angle of total external reflection may be related directly to the refractive index and therefore to
. Because the X-ray refractive indices of materials are only slightly less than unity, the scattering wavevector
is small, and the scattering angle is only a few degrees in magnitude. Assuming that there is not a strong dependence of
with
, one may consider that this technique provides an estimate of
for a photon energy range that cannot be surveyed using more precise techniques. A recent review of the use of reflectometers to determine
has been given by Lengeler (1994
).
This classification includes those experiments in which measurements of the geometrical structure factors for various Bragg reflections are undertaken. Into this category fall those techniques for which the period of standing-wave fields (Pendellösung) and reflectivity of perfect crystals in Laue or Bragg reflection are measured. Also included are those techniques from which the atomic scattering factors are inferred from measurements of Bijvoet- or Friedel-pair intensity ratios for noncentrosymmetric crystal structures.
The development of the dynamical theory of X-ray diffraction (see, for example, Part 5
in IT B, 2001
) and recent advances in techniques for crystal growth have enabled experimentalists to determine the geometrical structure factor
for a variety of materials by measuring the spacing between minima in the internal standing wave fields within the crystal (Pendellösung).
Two classes of Pendellösung experiment exist: those for which the ratio is kept constant and the thickness of the samples varies; and those for which the specimen thickness remains constant and
is allowed to vary.
Of the many experiments performed using the former technique, measurements by Aldred & Hart (1973a,b
) for silicon are thought to be the most accurate determinations of the atomic form factor
for that material. From these data, Price, Maslen & Mair (1978
) were able to refine values of
for a number of photon energies. Recently, Deutsch & Hart (1985
) were able to extend the determination of the form factor to higher values of momentum transfer
. This technique requires for its success the availability of large, strain-free crystals, which limits the range of materials that can be investigated.
A number of experimentalists have attempted to measure Pendellösung fringes for parallel-sided specimens illuminated by white radiation, usually from synchrotron-radiation sources. [See, for example, Hashimoto, Kozaki & Ohkawa (1965) and Aristov, Shmytko & Shulakov (1977
).] A technique in which the Pendellösung fringes are detected using a solid-state detector has been reported by Takama, Kobayashi & Sato (1982
). Using this technique, Takama and his co-workers have reported measurements for silicon (Takama, Iwasaki & Sato, 1980
), germanium (Takama & Sato, 1984
), copper (Takama & Sato, 1982
), and aluminium (Takama, Kobayashi & Sato, 1982
). A feature of this technique is that it can be used with small crystals, in contrast to the first technique in this section. However, it does not have the precision of that technique.
Another technique using the dynamical theory of X-ray diffraction determines the integrated reflectivity for a Bragg-case reflection that uses the expression for integrated reflectivity given by Zachariasen (1945). Using this approach, Freund (1975
) determined the value of the atomic scattering factor
for copper. Measurements of intensity are difficult to make, and this method is not capable of yielding results having the precisions of the Pendellösung techniques.
The Bijvoet-pair technique (Bijvoet et al., 1951) is used extensively by crystallographers to assist in the resolution of the phase problem in the solution of crystal structures. Measurements of as many as several hundred values for the diffracted intensities
for a crystal may be made. When these are analysed, the Cole & Stemple (1962
) observation that the ratio of the intensities scattered in the Bijvoet or Friedel pair is independent of the state of the crystal is assumed to hold. This is a necessary assumption since in a large number of structure analyses radiation damage occurs during the course of an experiment.
For simple crystal structures, Hosoya (1975) has outlined a number of ways in which values of
and
may be extracted from the Friedel-pair ratios. Measurements of these corrections for atoms such as gallium, indium, arsenic and selenium have been made.
In more complicated crystal structures for which the positional parameters are known, attempts have been made to determine the anomalous-scattering corrections by least-squares-refinement techniques. Measurements of these corrections for a number of atoms have been made, inter alia, by Engel & Sturm (1975), Templeton & Templeton (1978
), Philips, Templeton, Templeton & Hodgson (1978
), Templeton, Templeton, Philips & Hodgson (1980
), Philips & Hodgson (1985
), and Chapuis, Templeton & Templeton (1985
). There are a number of problems with this approach, not the least of which are the requirement to measure intensities accurately for a large period of time and the assumption that specimen perfection does not affect the intensity ratio. Also, factors such as crystal shape and primary and secondary extinction may adversely affect the ability to measure intensity ratios correctly. One problem that has to be addressed in this type of determination is the fact that
and
are related to one another, and cannot be refined separately.
In this section, discussion will be focused on (i) the scattering of photons having energies considerably greater than that of the K-absorption edge of the atom from which they are scattered, and (ii) scattering of photons having energies in the neighbourhood of the K-absorption edge of the atom from which they are scattered.
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
.
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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.
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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.
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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.
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