Soon after the invention of x-ray(x ray protection) diffraction it became dear that the method was not only suited to resolve crystalline structures, but could also reveal information on a sample's microstructure. In 1918 Scherrer devised a formula describing how the width of a Bragg reflection increases with decreasing crystallite size. The formula became very popular in the investigation of polycrystalline samples and made the crystallite size-peak width relation an ongoing issue in x-ray diffraction. It has to be emphasized, however, that the crystallite size derived from x-ray peak profiles accounts for domain sizes that scatter the incoming x-rays coherently. The size of coherently diffracting domains is generally smaller than the crystallite size as obtained by other techniques like transmission electron microscopy. It thus has to be carefully specified as to which experimental quantity is exactly meant whenever crystallite sizes are compared that have been obtained by different techniques.
The width of Bragg reflections was also realized to be affected by microstructural features other than size broadening, like the crystallite shape, the shape distribution and any distortion of the crystal structure like microstrain, dislocations,twin planes and stacking faults. The general question in the microstructural analysis of polycrystals relates to how the physical properties of the specimen are affected by its microstructure. It is evident that the microstructural richness of a polycrystalline thin film can become a hard task to be fully elucidated. The investigations are thus often restricted to certain aspects like the density of certain lattice faults.
Microstrain and dislocations play a prominent role in the various distortions of the crystal lattice. In the vicinity of dislocations the atoms reside on equilibrium po- sitions distinct from those in the unperturbed lattice and cause the surrounding bond lengths to contract and expand. The strain fields introduced by dislocations may extend over many hundreds of unit cells in the crystal. The interplanar spacing d as it appears in the Bragg equation is thus subjected to a variation and may not be accounted for by a fixed value do, but by a distribution of d values. The strain fields thus cause a smearing of scattered x-ray intensity around do and a broadening of reflections. These strain fields are denoted as microstrain, because they appear on a length scale that is small when compared with the inverse linear attenuation coefficient, 1/5, of the probing x ray beam. Although this definition is rather unsatisfactory, because a sample property is related to the technique by which it is probed, it is a practical approach and has become widely used. In consequence, the microstrain fields rise and fall within the x-ray illuminated sample volume. This contrasts with so-called macrostrains which exceed the 1/5 scale and can cause a complete shift of a Bragg reflection from to a new lattice spacing do + ed. In this chapter only the reflection broadening due to microstrains is considered, whereas peak shifts due to macrostrains are postponed to Chapter 6.
line profile analysis (LPA) endeavors to derive microstructural features in the sample from the shape and broadening of Bragg reflections. Because broadening due to crystallite size and microstrain typically occur together, techniques were to develop that allow for the separation of both effects. These developments were mainly performed investigations of bulk material from metals or simple in organic compounds. LPA ideally requires diffraction patterns with a high signal-to noise (S/N) ratio and freestanding reflections with negligible overlap. In many cases the usage of Kal monochromatized radiation is recommended to obtain reliable results. It is evident from this listing that the applications of LPA to thin films may not be straightforward. Some ofthe following examples will therefore relate to powder samples, for which the techniques were originally developed. It has to be emphasized that the crystallite orientation distribution is assumed in many LPA approaches to be a random one. This presupposition is not generally fulfilled in polycrystalline thin films: rather, preferred orientation or texture is typically ob- served. This does not invalidate the application of LPA(lead glass window) techniques in principle, but it should carefully be checked as to how far the interpretation of the data would be affected texture effects.
This chapter follows the historical development of LPA in so far as size effects are presented first, while the analysis of concomitant size and strain broadening is outlined subsequently. It should be kept in mind, however, that thin-film samples typically exhibit both microstructural properties. Instrumental Box 4 is devoted to numerical methods and software techniques that are used in the analysis of x-ray scattering data. Nanocrystalline and nanocomposite materials are chosen as the material class for which the concepts of LPA are illustrated. In these materials new functionalities may be elicited, since the ratio of surface over bulk atoms may achieve unusually large values. Structure Box 4 covers crystalline lattice faults.