Electron Diffraction and Crystallography
The practice of indexing single crystal diffraction patterns is explained, and worked in detail. Rules for generating and using stereographic projections for solving problems in crystallography are presented. Kikuchi lines are explained as a double-diffra
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Electron Diffraction and Crystallography
7.1 Indexing Diffraction Patterns Reciprocal lattices of crystals are spanned by three reciprocal-lattice vectors, so the diffraction patterns of materials are inherently three-dimensional. To obtain all available diffraction information, the diffraction intensity should be measured for B. Fultz, J. Howe, Transmission Electron Microscopy and Diffractometry of Materials, Graduate Texts in Physics, DOI 10.1007/978-3-642-29761-8_7, © Springer-Verlag Berlin Heidelberg 2013
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7 Electron Diffraction and Crystallography
all magnitudes and orientations of the three-dimensional diffraction vector, Δk. Appropriate spherical coordinates in k-space are Δk, θ , and φ. A practical approach to this fairly complicated problem is to separate the control over the magnitude of Δk, from its orientation with respect to the sample, θ and φ. The goniometers described in Sect. 1.3.2 provide the required control over the magnitude Δk, while on the sample. For isotropic polycrystalline maintaining a constant direction Δk samples, a single powder-diffraction pattern provides representative diffraction data because all crystal orientations are sampled. For specimens that are single crystals, however, it is also necessary to provide for the orientational degrees of freedom of the specimen (latitude and longitude angles, for example). A diffraction pattern (varying Δk) should then be obtained for each orientation within the selected solid angle (sin θ dθ dφ) of reciprocal space. Diffraction experiments with single crystals require additional equipment for specimen orientation, and software to relate these data to the reciprocal space structure of the three-dimensional crystal. For publication and display of these data, however, it is typical to present the diffraction intensities as planar sections through the three-dimensional data. Diffraction data from the TEM are obtained as near-planar sections through kspace. The large electron wavevector provides an Ewald sphere that is nearly flat, allowing the handy approximation that a diffraction pattern from a single crystal is a picture of a plane in its reciprocal space.1 The magnitude of the diffraction vector, Δk, is obtained from the angle between the transmitted and diffracted beams. Two degrees of orientational freedom are required for the sample in a TEM. They are typically provided by a “double-tilt specimen holder,” which has two perpendicular tilt axes oriented perpendicular to the incident electron beam. A modern TEM provides two modes for obtaining diffraction patterns from individual crystallites. The oldest is selected area diffraction (SAD), which is useful for obtaining diffraction patterns from regions as small as 0.5 µm in diameter (see Problem 2.16). The second method is nanodiffraction, or convergent-beam electron diffraction (CBED), in which a focused electron probe beam is used to obtain diffraction patterns from regions as small as 10 Å. Both techniques provide a two-dimensional pattern of diffraction spots, which can be
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