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

International Tables for Crystallography (2006). Vol. C, ch. 6.4, p. 609

Section 6.4.2.  The model of a real crystal

T. M. Sabinea

aANSTO, Private Mail Bag 1, Menai, NSW 2234, Australia

6.4.2. The model of a real crystal

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Following Darwin (1922[link]), a crystal is assumed to consist of small blocks of perfect crystal (called mosaic blocks). These blocks are separated by small-angle boundaries formed by dislocations (Read, 1953[link]), which introduce random tilts of the blocks with respect to each other. It is necessary to divide this model into two subclasses. In the first, the introduction of tilts does not destroy the spatial correlation between atoms in different blocks that have the same relative orientation. In this model (hereafter called the correlated block model), the removal of dislocations constituting the mosaic block boundaries by, for example, thermal annealing will recover a monolithic perfect single crystal. In the second subclass, which is the original Darwin model (hereafter called the uncorrelated block model), the introduction of tilts destroys spatial coherence between blocks. The weakness of the latter model is that removal of the tilts does not recover the monolithic single crystal, but leads to a brick-wall-type structure with parallel bricks but varying thicknesses of mortar between the bricks. In practice, thermal-annealing treatments at sufficiently high temperatures always regain the single crystal.

The existence of this problem for extinction theory has been recognized (Wilkins, 1981[link]). Theories will be given in this article for both limits. For a more complete treatment, the introduction of a mixing parameter proportional to either the dislocation density in the crystal or the physical proximity of mosaic blocks may be necessary.

For a single incident ray, only those blocks whose orientation is within the natural width for Bragg scattering from a perfect crystal will contribute to Bragg diffraction processes. For these processes, the crystal presents a sponge- or honeycomb-like aspect, since blocks that are outside the angular range for scattering do not contribute to the diffraction pattern. During the scan used for measurement of the integrated intensity, all blocks will contribute. At any angular setting, the entire volume of the crystal contributes to absorption and other non-Bragg-type interactions.


Darwin, C. G. (1922). The reflexion of X-rays from imperfect crystals. Philos. Mag. 43, 800–829.
Read, W. T. (1953). Dislocations in crystals. New York: McGraw-Hill.
Wilkins, S. W. (1981). Dynamical X-ray diffraction from imperfect crystals in the Bragg case – extinction and the asymmetric limits. Philos. Trans. R. Soc. London, 299, 275–317.

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