Electron Diffraction and Crystallography

Abstract

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 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. The goniometers described in Sect. 1.3.2 provide the required control over the magnitude Δk, while maintaining a constant direction $\widehat{\Delta k}$ on the sample. For isotropic polycrystalline 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 (sinθ dθdφ) 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.