International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C. ch. 2.7, pp. 117119

The foregoing description of singlecrystal techniques will have indicated that in order to gain greater sensitivity in orientation contrast there are required incident beams with closer collimation, and limitation of dispersion due to wavelength spread of the characteristic Xray lines used. It suggests turning to prior reflection of the incident beam by a perfect crystal as a means of meeting these needs. Moreover, the application of crystalreflectioncollimated radiation to probe angularly step by step as well as spatially point by point the intensity of Bragg reflection from the vicinity of an individual lattice defect such as a dislocation brings possibilities of new measurements beyond the scope provided by simply recording the local value of the integrated reflection. The Xray optical principles of doublecrystal Xray topography are basically those of the doublecrystal spectrometer (Compton & Allison, 1935). The properties of successive Bragg reflection by two or more crystals can be effectively displayed by a Du Mond diagram (Du Mond, 1937), and such will now be applied to show how collimation and monochromatization result from successive reflection by two crystals, U and V, arranged as sketched in Fig. 2.7.3.1 . They are in the dispersive, antiparallel, `+ +' setting, and are assumed to be identical perfect crystals set for the same symmetrical Bragg reflection. Only rays making the same glancing angle with both surfaces will be reflected by both U and V. For example, radiation of shorter wavelength reflected at a smaller glancing angle at U (the ray shown by the dashed line) will impinge at a larger glancing angle on V and not satisfy the Bragg condition. In this + + setting, with a given angle ω between the Braggreflecting planes of each crystal, = ω and . The Du Mond diagram for the + + setting, Fig. 2.7.3.2 , shows plots of Bragg's law for each crystal, the V curve being a reflection of the U curve in a vertical mirror line and differing by ω from the U curve in its coordinate of intersection with the axis of abscissa, in accord with the equations given above. The small angular range of reflection of a monochromatic ray by each perfect crystal is represented exaggeratedly by the band between the parallel curves. Where the band for crystal U superimposes on the band for V (the shaded area) defines semiquantitatively the divergence and wavelength spread in the rays successively reflected by U and V. (It is taken for granted that lies between the maximum and minimum incident glancing angles on U, and , afforded by the incident beam, assumed polychromatic.) The reflected beam from U alone contains wavelengths ranging from to . Comparison of these and ranges with the extent of the shaded area illustrates the efficacy of the + + arrangement in providing a collimated and monochromatic beam, which can be employed to probe the reflecting properties of a third crystal (Nakayama, Hashizume, Miyoshi, Kikuta & Kohra, 1973). Techniques employing three or more successive Bragg reflections find considerable application when used with synchrotron Xray sources, and will be considered below, in Section 2.7.4.
The most commonly used arrangement for doublecrystal topography is shown in Fig. 2.7.3.3 , in which U is the `reference' crystal, assumed perfect, and V is the specimen crystal under examination. Crystals U and V are in the parallel, `+ −' setting, which is nondispersive when the Bragg planes of U and V have the same (or closely similar) spacings. Before considering the Du Mond diagram for this arrangement, note that Bragg reflection at the reference crystal U is asymmetric, from planes inclined at angle α to its surface. Asymmetric reflections have useful properties, discussed, for example, by Renninger (1961), Kohra (1972), Kuriyama & Boettinger (1976), and Boettinger, Burdette & Kuriyama (1979). The asymmetry factor, b, of magnitude , n being the crystalsurface normal, is also the ratio of spatial widths of the incoming to the outgoing beams, W_{in}/W_{out}. In the case of symmetric Bragg reflection, the perfect crystal U would totally reflect (in the zeroabsorption case) over a small angular range, . In the asymmetric case, the ranges of total reflection are w_{in} for the incoming rays and w_{out} for the outgoing. Dynamicaldiffraction theory [IT B (2001, Part 5 )] shows that , so that w_{in}W_{in} = w_{out}W_{out} (as would be expected from energy conservation). Thus, highly asymmetric reflection from the reference crystal U not only provides a spatially wide beam, able to cover a large area of V without recourse to any mechanical traversing motion of the components S, U or V, but also produces a desirably narrow angular probe for studying the angular breadth of reflection of V. In practice, values of b lower than 0.1 can be used.

Doublecrystal topographic arrangement, + − setting. Asymmetric reflection from reference crystal U. Specimen crystal divided into regions V and V′. 
Du Mond diagrams for the + − arrangement are shown in Fig. 2.7.3.4 (a) and (b). For simplicity, the curves (slope ) are represented by straight lines. In the + − setting, and . In Fig. 2.7.3.4(a), the narrow band labelled U passing through the origin represents the beam of angular width w_{out} leaving U. It is assumed that all of the specimen crystal V has the same interplanar spacing as U but that it contains a slightly misoriented minor region V′ (which may be located as shown in Fig. 2.7.3.3). When ω differs substantially from zero, the bands corresponding to crystal V and its minor part V′ lie in positions and , respectively, in Fig. 2.7.3.4(a). (Only the relevant part of the latter band is drawn, for simplicity.) The offset along the axis between and is the component of the misorientation between V and V′ that lies in the plane of incidence. If ω is reduced stepwise, a doublecrystal topograph image being obtained at F at each angular setting, can be found from film densitometry, which will show at what settings band U is most effectively overlapped by band V or by band V′. When ω is reduced to zero, the specimen crystal bands are at and . The drawing shows that V′ has then passed right through the setting for its Bragg reflection, which occurred at a small positive value of ω. Since the U and V bands have identical slopes, their overlap occurs at all wavelengths when ω = 0. In practice, only the shaded area is involved, corresponding to the wavelength range to , defined by the range of incidence angles, to , on the Bragg planes of crystal U. (The width of band U will generally be negligible compared with the range of allowed by source width and slit collimation system.) One component of is found in the procedure just described. The second component needed to specify the difference between hvector directions of the Bragg planes of V and V′ is obtained by repeating the experiment after rotating V by 90° about h.
Next consider the more general case when V′ differs from V in both orientation and interplanar spacing, and both V′ and V have slightly different interplanar spacings from U. The difference in orientation between V′ and V, , and their difference in interplanar spacing, d(V′) − d(V), can be distinguished by taking two series of doublecrystal topographs, the orientation of the specimen in its own plane (its azimuthal angle, ψ) being changed by a 180° rotation about its h vector between taking the first and second series. As shown schematically in the Du Mond diagram, Fig. 2.7.3.4(b), the U, V, and V′ bands now all have slightly different slopes. [Reference crystal U is reflecting the same small wavelength band as in Fig. 2.7.3.4(a).] The setting represented in the diagram is that putting V at the maximum of its Bragg reflection Let the V′ band be then at position , for the case when ψ = 0°. Assume that, when ψ is changed by 180°, the rotation of the specimen in its own plane can be made about the h vector of V precisely. (This assumption simplifies the diagram.) Then this 180° rotation will not cause any translation of the V band along the axis, but does transfer the V′ band from to the position . With the sense of increasing ω taken as that translating the specimen bands to the right and taken as the difference in readings between peak reflection from V and that from V′, the diagram shows that, with , , and, with ψ = 180°, , from which both and the difference in Bragg angles can be found. The interplanar spacing difference is given by , being the mean interplanar spacing of V and V′. In practice, series of topographs are taken with azimuthal angles ψ = 0, 90, 180, and 270°, so that the two components needed to specify the misorientation vector between the Braggplane normals of V and V′ can be determined. The Du Mond diagram shows that in this slightly dispersive experiment the range of overlap of the U band with any V band can be restricted by reducing the angular range or wavelength range of rays incident on U. Such reduction can be achieved by use of a small source S far distant from U, such as a synchrotron source. It can also be achieved by methods described in Subsection 2.7.4.2. As regards spatial resolution on doublecrystal topographs, relations analogous to those for singlecrystal topographs apply. If the reference crystal U unavoidably contains some defects, their images on F can deliberately be made diffuse compared with images of defects in V by making the UV distance relatively large. In a nearly dispersionfree arrangement, if the wavelength is being reflected, then so too will the if S is sufficiently widely extended in the incidence plane, as is usually necessary to image a usefully large area of V. If the distance VF cannot be made sufficiently small to reduce to a tolerable value the resolution loss due to simultaneous registration of the and images, then a source S of small apparent size, together with a collimating slit before U, will be needed. In order to obtain imaging of a large area of V, a linear scanning motion to and fro at an angle to SU in the plane of incidence must be performed by the source and collimator relative to the doublecrystal camera. Whether it is the source and collimator or the camera that physically move depends upon their relative portability. When the source is a standard sealedoff Xray tube, it is not difficult to arrange for it to execute the motion (Milne, 1971).
In some applications, it may occur that the specimen is so deformed that only a narrow strip of its surface will reflect at each ω setting. Then, a sequence of images can be superimposed on a single film, changing ω by a small step between each exposure. The `zebra' patterns so obtained define contours of equal `effective misorientation', the latter term describing the combined effect of variations in tilt and of Braggangle changes due to variations in interplanar spacing (Renninger, 1965; Jacobs & Hart, 1977).
Doublecrystal topography employing the parallel setting was developed independently by Bond & Andrus (1952) and by Bonse & Kappler (1958), and used by the former workers for studying reflections from surfaces of natural quartz crystals, and by the latter for detecting the strain fields surrounding outcrops of single dislocations at the surfaces of germanium crystals. Since then, the method has been much refined and widely applied. The detection of relative changes in interplanar spacing with a sensitivity of 10^{−8} is achievable using highangle reflections and very perfect crystals. These developments have been reviewed by Hart (1968, 1981).
Transmitted Bragg reflection (i.e. the Laue case) can be used for either or both crystals U and V, in both the + + and + − settings, if desired. When the reference crystal U is used in transmission, a technique due to Chikawa & Austerman (1968), shown in Fig. 2.7.3.5 , can be employed if U is relatively thick and, preferably, not highly absorbing of the radiation used. This technique exploits a property of diffraction by ideally perfect crystals, that, for waves satisfying the Bragg condition exactly, the energyflow vector (Poynting vector) within the energyflow triangle (the triangle ORT in Figs. 2.7.2.2 and 2.7.2.3) is parallel to the Bragg planes. (In fact, the energyflow vectors swing through the triangle ORT as the range of Bragg reflection is swept by the incident wave vector, .) Placing a slit Q as shown in Fig. 2.7.3.5 so as to transmit only those diffracted rays emerging from RT whose energyflow direction in the crystal ran parallel, or nearly parallel, to the Bragg plane OD has the effect of selecting out from all diffracted rays only those that have zero or very small angular deviation from the exact Bragg condition. The slit Q thus provides an angularly narrower beam for studying the specimen crystal V than would be obtained if all diffracted rays from U were allowed to fall on V. The specimen is shown here in the setting, and also oriented to transmit its diffracted beam to the film F. This specimen arrangement is a likely embodiment of the technique but is incidental to the principle of employing spatial selection of transmitted diffraction rays to gain angular selection, a technique first used by Authier (1961). A practical limitation of this technique arises from angular spreading due to Fraunhofer diffraction by the slit Q: use of too fine an opening of Q will defeat the aim of securing an extremely angularly narrow beam for probing the specimen crystal.
References
Authier, A. (1961). Etude de la transmission anormale des rayons X dans des cristaux de silicium. I. Case des cristaux parfaits. Bull. Soc. Fr. Minéral. Cristallogr. 84, 51–89.Google ScholarBoettinger, W. J., Burdette, H. E. & Kuriyama, M. (1979). Xray magnifier. Rev. Sci. Instrum. 50, 26–30.Google Scholar
Bond, W. L. & Andrus, J. (1952). Structural imperfections in quartz crystals. Am. Mineral. 37, 622–632.Google Scholar
Bonse, U. & Kappler, E. (1958). Röntgenographische Abbildung des Verzerrungsfeldes einzelner Versetzungen in GermaniumEinkristallen. Z. Naturforsch. Teil A, 13, 348–349.Google Scholar
Chikawa, J.I. & Austerman, S. B. (1968). Xray doublecrystal method of analyzing microstrains with BeO single crystals. Advances in Xray analysis, Vol. 11, edited by J. B. Newkirk & G. R. Mallett, pp. 393–400. New York: Plenum.Google Scholar
Compton, A. H. & Allison, S. K. (1935). Xrays in theory and experiment. New York: Van Nostrand.Google Scholar
Gerold, V. & Meier, F. (1959). Der röntgenographische Nachweiss von Versetzungen in Germanium. Z. Phys. 155, 387–394.Google Scholar
Hart, M. (1968). `Perfect crystals'. A study of their structural defects. Sci. Prog. Oxford, 56, 429–447. Google Scholar
Hart, M. (1981). Bragg angle measurement and mapping. J. Cryst. Growth, 55, 409–427.Google Scholar
International Tables for Crystallography (2001). Vol. B. Dordrecht: Kluwer Academic Publishers.Google Scholar
Jacobs, L. & Hart, M. (1977). An Xray topographic study of large crystals for a bentcrystal gamma diffractometer. Nucl. Instrum. Methods, 143, 319–325.Google Scholar
Kohra, K. (1972). Dynamical asymmetric diffraction and its applications to Xray optical systems. Proceedings of the VIth International Conference on Xray Optics and Microanalysis, edited by G. Shinoda, K. Kohra & T. Ichinokawa, pp. 35–45. Tokyo: University of Tokyo Press.Google Scholar
Kuriyama, M. & Boettinger, W. J. (1976). On the angular divergence of outgoing beams in an asymmetric diffraction geometry. Acta Cryst. A32, 511–512.Google Scholar
Milne, A. D. (1971). Scanning source Xray topography. J. Appl. Cryst. 4, 251–252. Google Scholar
Nakayama, K., Hashizume, H., Miyoshi, A., Kikuta, S. & Kohra, K. (1973). Use of asymmetrical dynamical diffraction of Xrays for multiplecrystal arrangements of the (n_{1}, +n_{2}) setting. Z. Naturforsch. Teil. A, 28, 632–638.Google Scholar
Renninger, M. (1961). Asymmetrische BraggReflexion am Idealkrystall zur Erhöhung des DoppelspekrometerAuflösungsvermögens. Z. Naturforsch. Teil. A, 16, 1110–1111.Google Scholar
Renninger, M. (1965). Beitrage zur doppeldiffraktometrischen KristallTopographic mit Röntgenstrahlen I. Methodik und Ergebnisse typischer Art. Z. Angew. Phys. 19, 20–33.Google Scholar