Determination of the real part of the dispersion correction:

Creagh, D. C.

doi:10.1107/97809553602060000592

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

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International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 250-251

Section 4.2.6.3.2. Determination of the real part of the dispersion correction: $[f'(\omega,{\boldDelta})]$

D. C. Creagh^b

4.2.6.3.2. Determination of the real part of the dispersion correction: $[f'(\omega,{\boldDelta})]$

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This classification includes those experiments in which measurements of the geometrical structure factors $[F_{hkl}]$ 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.

4.2.6.3.2.1. Measurements using the dynamical theory of X-ray diffraction

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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 $[F_{hkl}]$ 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 $[(\lambda/\cos\theta)]$ is kept constant and the thickness of the samples varies; and those for which the specimen thickness remains constant and $[(\lambda/\cos\theta)]$ 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 $[f(\omega,{\boldDelta})]$ for that material. From these data, Price, Maslen & Mair (1978) were able to refine values of $[f'(\omega,{\boldDelta})]$ 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 $[(\hbar{\boldDelta})]$ . 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 $[f(\omega,{\bf g}_{222})]$ 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.

4.2.6.3.2.2. Friedel- and Bijvoet-pair techniques

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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 $[I_{hkl}]$ 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 $[f'(\omega,{\bf g}_{hkl})]$ and $[f''(\omega,{\bf g}_{hkl})]$ 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 $[f'(\omega,0)]$ and $[f''(\omega,0)]$ are related to one another, and cannot be refined separately.

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International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 250-251