Measurements using the dynamical theory of X-ray diffraction

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

D. C. Creagh^b

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.

References

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