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

International Tables for Crystallography (2006). Vol. C, ch. 2.7, pp. 121-122

Section Moiré topography

A. R. Langa

aH. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, England Moiré topography

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In X-ray optics, the same basic geometrical interpretation of moiré patterns applies as in light and electron optics. Suppose radiation passes successively through two periodic media, (1) and (2), whose reciprocal vectors are [{\bf h}_1] and [{\bf h}_2], so as to form a moiré pattern. Then, the reciprocal vector of the moiré fringes will be [{\bf H}={\bf h}_1-{\bf h}_2]. The magnitude, D, of the moiré fringe spacing is [|{\bf H}|^{-1}] and may typically lie in the range 0.1 to 1 mm in the case of X-ray moiré patterns. Simple special cases are the `rotation' moiré pattern in which [|{\bf h}_1|=|{\bf h}_2|=d^{-1}], but [{\bf h}_1] makes a small angle α with [{\bf h}_2]. Then, the spacing of the moiré fringes is d/α and the fringes run parallel to the bisector of the small angle α. The other special case is the `compression' moiré pattern. Here, [{\bf h}_1] and [{\bf h}_2] are parallel but there is a small difference between their corresponding spacings, [d_1] and [d_2]. The spacing D of compression moiré fringes is given by D = d1d2/(d1d2) and the fringes lie parallel to the grating rulings or Bragg planes in (1) and (2). X-ray moiré topographs achieve sensitivities of [10^{-7}] to [10^{-8}] in measuring orientation differences or relative differences in interplanar spacing. Moreover, if either periodic medium contains a lattice dislocation, Burgers vector b, for which [{\bf b}\cdot{\bf h}\ne0], then a magnified image of the dislocation will appear in the moiré pattern, as one or more fringes terminating at the position of the dislocation, the number of terminating fringes being [{\bf b}\cdot{\bf h}], which is necessarily integral (Hashimoto & Uyeda, 1957[link]).

X-ray moiré topography has been performed with two quite different arrangements, the Bonse & Hart interferometer, and by superposition of separate crystals (Brádler & Lang, 1968[link]). For accounts of the principles and applications of the interferometer, see, for example, Bonse & Hart (1965b[link], 1966[link]), Hart (1968[link], 1975b[link]), Bonse & Graeff (1977[link]), Section 4.2.6[link] and §[link] . Fig.[link] shows the arrangement (Hart, 1968[link], 1972[link]) for obtaining large-area moiré topographs by traversing the interferometer relative to a ribbon incident beam in similar fashion to taking a normal projection topograph (Fig.[link]); P is the incident-beam slit, Q is a stationary slit selecting the beam that it is desired to record, and film, F, and interferometer, SMA, together traverse to and fro as indicated by the double-headed arrow. In Fig.[link], S, M, and A are the three equally thick wafers of the interferometer that remain upstanding above the base of the monolithic interferometer after the gaps between S and M, and M and A, have been milled away. The elements S, M and A are called the splitter, mirror, and analyser, respectively. The moiré pattern is formed between the Bragg planes of A and the standing-wave pattern in the overlapping [{\bf K}_0] and [{\bf K}_h] beams entering it. Maximum fringe visibility occurs in the emerging beam that the slit Q is shown selecting. A dislocation will appear in the moiré pattern whether the lattice dislocation lies in S, M, or A, provided [{\bf b}\cdot{\bf h}\ne0]. Moiré patterns formed in a number of Bragg reflections whose normals lie in, or not greatly inclined to, the plane of the wafers, can be recorded by appropriate orientation of the monolith. By this means, it is easily discovered in which wafer the dislocation lies, and its Burgers vector can be completely determined, including its sense, the latter being found by a deliberate slight elastic deformation of the interferometer (Hart, 1972[link]). Satisfactory moiré topographs have been obtained with an interferometer in a synchrotron beam, despite thermal gradients due to the local intense irradiation (Hart, Sauvage & Siddons, 1980[link]).


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Scanning arrangement for moiré topography with the Bonse–Hart interferometer.

Fig.[link] shows crystal slices (1), ABCD, and (2), EFGH, superposed and simultaneously Bragg reflecting in the Brádler–Lang (1968[link]) method of X-ray moiré topography. The slices could have been cut from separate crystals. In the case when the Bragg planes of (1) and (2) are in identical orientation but have a translational mismatch across CD and EF with a component parallel to h, strong scattering occurs towards Z as focus, producing extra intensity at [T'] in the [{\bf K}_0] beam [TT''] and at [R'] in the [{\bf K}_h] beam [RR'']. It is usual to record the moiré pattern using the [{\bf K}_h] beam. Projection moiré topographs are obtained by the standard procedure of traversing the crystal pair and film together with respect to the incident beam SO. The special procedure devised for mutually aligning the two crystals so that [{\bf h}_1] and [{\bf h}_2] coincide within their angular range of reflection is explained by Brádler & Lang (1968[link]). This method has been applied to silicon and to natural (Lang, 1968[link]) and synthetic quartz (Lang, 1978[link]).


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Superposition of crystals (1) and (2) for production of moiré topographs. [Reproduced from Diffraction and Imaging Techniques in Material Science, Vol. II. Imaging and Diffraction Techniques, edited by S. Amelinckx, R. Gevers & J. Van Landuyt (1978), Fig. 21, p. 695. Amsterdam, New York, Oxford: North-Holland.]


Bonse, U. & Graeff, W. (1977). X-ray and neutron interferometry. X-ray optics. Applications to solids, edited by H.-J. Queisser, Chap. 4, pp. 93–143. Berlin: Springer.
Bonse, U. & Hart, M. (1965b). An X-ray interferometer. Appl. Phys. Lett. 6, 155–158.
Bonse, U. & Hart, M. (1966). Moiré patterns of atomic planes obtained by X-ray interferometry. Z. Phys. 190, 455–467.
Brádler, J. & Lang, A. R. (1968). Use of the Ewald sphere in aligning crystal pairs to produce X-ray moiré fringes. Acta Cryst. A24, 246–247.
Hart, M. (1968). `Perfect crystals'. A study of their structural defects. Sci. Prog. Oxford, 56, 429–447.
Hart, M. (1972). A complete determination of dislocation Burgers vectors by X-ray interferometry. Philos. Mag. 26, 821–831.
Hart, M. (1975b). Ten years of X-ray interferometry. Proc. R. Soc. London Ser. A, 346, 1–22.
Hart, M., Sauvage, M. & Siddons, D. P. (1980). `White beam' synchrotron X-ray interferometry. Acta Cryst. A36, 947–951.
Hashimoto, H. & Uyeda, R. (1957). Detection of dislocation by the moiré pattern in electron micrographs. Acta Cryst. 10, 143.
Lang, A. R. (1968). X-ray moiré topography of lattice defects in quartz. Nature (London), 220, 652–657.
Lang, A. R. (1978). Techniques and interpretation in X-ray topography. Diffraction and imaging techniques in material science, 2nd, revised edition, edited by S. Amelinckx, R. Gevers & J. Van Landuyt, pp. 623–714. Amsterdam: North-Holland.

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