ABSTRACT
In an ideal crystal lattice the environment of all points is identical, but only in exceptional circumstances does this occur in practice; "real" materials contain structural imperfections, known as microstructure, which broaden diffraction line profiles and, in some cases, give rise to displacement of reflections and introduce symmetry. In angle-dispersive (fixedwavelength) experiments, a one-dimensional variation of intensity with scattering angle 28 is obtained, but by combining data from several reflections, including as many higher orders as are available, a three-dimensional model of the sample microstructure can in principle be constructed. The same is true of energy-dispersive studies, where 28 is fixed and the incident radiation contains a range of wavelengths. In order to interpret the diffraction effects of sample imperfections from either type of experiment and derive information on microstructure, it is convenient to use the reciprocal-lattice representation. Imperfections give rise to a spread of intensity around the reciprocal-lattice point corresponding to each reflection and data from a random powder can then be regarded as a one-dimensional variation of intensity with the radial direction dhkl , where
Structural imperfections that modifY diffraction line profiles have in the past been broadly classified as "size effects," "strain," and "mistakes," and this phenomenological approach is followed here. Size effects arise from the finite thickness, in the direction dhkl , of domains over which diffraction is coherent. This can be the thickness of individual crystallites (or grains, in a polycrystalline sample), but it can also relate to a subdomain structure, such as the mean distance between structural "mistakes," the separation of regions bounded by low-angle grain boundaries, etc. Here, unless specified otherwise, size or crystallite size is used to refer to any of these effects. The second category is based on a distortion of the crystal lattice, which amounts to a variation of d spacing within (or possibly between) domains. This can arise from microstrain, due to an applied or internal stress, or from a compositional gradient in the sample. It is not possible to distinguish between these
effects from diffraction data, but the distinction may be evident from the nature of the sample. For convenience, they are known as "strain effects." Dislocations contribute to both categories ofline broadening; there will be a size contribution due to their mean separation, inversely proportional to the dislocation density, and microstrain arising from internal stress fields. If structural "mistakes" are present, the arrangement of atoms differs in different regions of a crystallite. This can arise during crystallization, or it can be induced by some external influence, such as heat treatment, plastic deformation, adsorption, or radiation damage. Strain can be expressed as /1.d/d, and the corresponding line broadening thus increases linearly with the order of a reflection. Size and mistake broadening, on the other hand, do not depend directly Idhk,l, but the former can vary with lattice direction if the shape of domains is other than spherical and the latter depends on variations in structure factor and hence on hkl. These properties can be used to identifY and separate the contributions to the overall breadth arising from structural imperfections (see Section 3.4), though an alternative procedure based on a Fourier approach, which avoids assumptions regarding the order dependence of diffraction effects, is discussed in Chapter 42.
