International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 248-250

Section 4.2.6.3.1. Determination of the real part of the dispersion correction: [f'(\omega,0)]

D. C. Creaghb

4.2.6.3.1. Determination of the real part of the dispersion correction: [f'(\omega,0)]

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X-ray interferometer techniques are now used extensively for the measurement of the refractive index of materials and hence [f'(\omega,0)]. All the interferometers are transmission-geometry LLL devices (Bonse & Hart, 1965b[link], 1966a[link],b[link],c[link],d[link], 1970[link]), 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 [K\alpha_1], Mo [K\alpha_1], Cu [K\alpha_1] and Cr [K\alpha_1] by a variety of authors (Creagh & Hart, 1970[link]; Creagh, 1970[link]; Bonse & Hellkötter, 1969[link]; Bonse & Materlik, 1972[link]).

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[link]) 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[link]). 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[link]) 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[link], 1977[link]). Subsequently, measurements have been made by Siddons & Hart (1983[link]) and Hart & Siddons (1981[link]) for zirconium, niobium, nickel, and molybdenum. Hart (1985[link]) 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 [f'(\omega,0)] 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[link]).

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 [f'(\omega,0)].

Deutsch & Hart (1984a[link],b[link]) 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[link]) made determinations of [f'(\omega,0)] in the region of the K-absorption edge for copper. More recently, Ishida & Katoh (1982[link]) have described the use of a multiple-reflection diffractometer for the determination of X-ray refractive indices. Later, Katoh et al. (1985a[link],b[link]) described its use for the measurement of [f'(\omega,0)] for lithium fluoride and potassium chloride at a wavelength near that of Mo [K\alpha_1] and for germanium in the neighbourhood of its K-absorption edge.

Measurements of the linear attenuation coefficient [\mu_l] over an extended energy range can be used as a basis for the determination of the real part of the dispersion correction [f'(\omega,0)] because of the Kramers–Kronig relation, which links [f'(\omega,0)] and [f''(\omega,0)]. However, as Creagh (1980[link]) has pointed out, even if the integration can be performed accurately [implying the knowledge of [f''(\omega,0)] 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[link]. 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 [f'(\omega,0)] occur for very small changes in photon energy [\hbar\omega].

Despite these difficulties, Creagh (1977[link], 1978[link], 1982[link]) has used the technique to determine [f'(\omega,0)] and [f''(\omega,0)] for several alkali halides and Gerward, Thuesen, Stibius-Jensen & Alstrup (1979[link]) used the technique to measure these dispersion corrections for germanium. More recently, the technique has been used by Dreier et al. (1984[link]) to determine [f'(\omega,0)] and [f''(\omega,0)] 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[link]) to study a variety of amorphous materials in the region of an absorption edge.

Henke et al. (1982[link]) 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 [(\hbar\omega\lt1.5\,{\rm keV})].

Linear attenuation coefficient measurements yield [f'(\omega,0)] directly and [f''(\omega,0)] 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[link]) 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 [f'(\omega,{\boldDelta})]. Because the X-ray refractive indices of materials are only slightly less than unity, the scattering wavevector [{\boldDelta}] is small, and the scattering angle is only a few degrees in magnitude. Assuming that there is not a strong dependence of [f'(\omega,{\boldDelta})] with [{\boldDelta}], one may consider that this technique provides an estimate of [f'(\omega,0)] for a photon energy range that cannot be surveyed using more precise techniques. A recent review of the use of reflectometers to determine [f'(\omega,0)] has been given by Lengeler (1994[link]).

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