Line Profiles Caused by Planar Faults

The planar faults in crystalline materials yield characteristic broadening of X-ray line profiles. The diffraction peak shape caused by intrinsic and extrinsic stacking faults and twin boundaries formed on close packed {111} planes in face centered cubic (fcc) crystals are calculated. The Bragg reflections consist of subreflections that can be categorized by specific selection rules for the hkl indices. The breadth and the position of the subreflections relative to the exact Bragg angle depend on their indices. For instance, if the sum of indices of a subreflection is a multiple of three, neither the position nor the breadth of this peak is influenced by planar faults. Other subreflections are broadened and shifted simultaneously due to intrinsic and extrinsic stacking faults. For both fcc and hexagonal close packed (hcp) crystals each subreflection caused by twin boundaries is a sum of symmetric and antisymmetric Lorentzian functions. The latter profile component is caused by the interference between the radiations scattered from the parent and twinned lamellae in the crystal. The antisymmetric Lorentzian function yields a shift of the subprofile center. For fcc materials this displacement of peak position is marginal since twin boundaries are formed on close packed {111} planes; however in hcp crystals, where twinning usually occurs on pyramidal planes, this effect should be taken into account in the line profile evaluation. The effect of anti-phase boundaries on line profiles of superstructure reflections for Cu3Au is also discussed in this chapter.

Fundamentals of Kinematical X-Ray Scattering Theory

The broadening of X-ray line profiles is usually described by the kinematical scattering theory. In this chapter, the basic concepts and equations of the kinematical X-ray scattering are presented in order to better understand the theory of line profile analysis. The correlation between the crystal structure and the diffracted intensity distribution is shown. The scattering angles of the diffracted peak maxima are given by the Ewald construction in the reciprocal space. The correspondence between the reciprocal lattice vectors and the lattice planes is also presented, and the relationship between the scattering angle and the lattice plane spacing is given by Bragg’s law.

Practical Applications of X-Ray Line Profile Analysis

In the previous chapters, the theory and the main methods of diffraction peak profile analysis were presented. Additionally, the specialties in the measurement and the evaluation of line profiles in the cases of thin films and single crystals were discussed. In this chapter, some practical considerations are given in order to facilitate the evaluation of peak profiles and the interpretation of the results obtained by this method. For instance, the procedures for instrumental correction are overviewed. Additionally, how the prevailing dislocation slip systems and twin boundary types in hexagonal polycrystals can be determined from line profiles is shown. Besides the dislocation density, the vacancy concentration can also be obtained by the combination of electrical resistivity, calorimetric, and line profile measurements. The crystallite size and the twin boundary frequency determined by X-ray peak profile analysis are compared with the values obtained by the direct method of transmission electron microscopy. Furthermore, the limits of line profile analysis in the determination of crystallite size and defect densities are given. Finally, short overviews on the results obtained by peak profile analysis for metals, ceramics, and polymers are presented.

Peak Profile Evaluation for Thin Films

The special phenomena in X-ray diffraction line profile analysis occurring in thin films is overviewed in this chapter. In the case of textured nanocrystalline thin films, the line broadening caused by the crystallite size increases with the length of the diffraction vector. This effect is explained by the interference of X-rays scattered coherently from adjacent crystallites with close orientations. The partial coherence of adjacent nanocrystallites is caused by the overlapping of their reciprocal lattice points. The smaller the size and the stronger the orientation preference of crystallites, the better the coherence. This interference effect yields narrowing of line profiles at small diffraction angles, while it has no influence on line broadening at large angles. Therefore, the traditional line profile evaluation methods give much larger crystallite size than the real value and may detect a false microstrain broadening. Some ways for the correction of the interference effect are proposed. Detailed case studies are given for the determination of the defect structure in thin films by line profile analysis.

X-Ray Line Profile Analysis for Single Crystals

The features of the dislocation structure in plastically deformed single crystals can be determined from diffraction line broadening. Both the measuring and the evaluation procedures of X-ray line profiles are somewhat different from the methods used for polycrystalline materials. In this chapter, these procedures are overviewed, and their effectiveness is illustrated by representative examples. It is shown that the intensity distribution in the vicinity of the reciprocal lattice points can be mapped by rocking the single crystal about appropriate axes. From the detected intensity distribution, the density, the slip systems, and the arrangement of dislocations, as well as the lattice misorientation can be determined. The average misorientation obtained from rocking curve measurement can be related to the density of geometrically necessary dislocations. It is also shown that the inhomogeneous distribution of dislocations in plastically deformed single crystals usually results in asymmetric line profiles. The evaluation of these peaks enables the determination of the long-range internal stresses besides the dislocation densities in the dislocation cell walls and interiors.

Strain Broadening of X-Ray Diffraction Peaks

The line shape caused by lattice distortions in a crystal is reviewed. It is revealed that the broadening of a diffraction peak with indices hkl is related to the mean-square-strain perpendicular to the reflecting (hkl) lattice planes. The strain broadening of line profiles depends on the order of diffraction. The line profiles for a crystal in which the lattice distortions are caused by dislocations are described in detail in this chapter. It is revealed that the anisotropic strain field of dislocations yields a special dependence of peak broadening on indices of reflection. The stronger the screening of the strain fields of dislocations, the longer the tails in the diffraction profiles. For polarized dislocation walls, the diffraction peak is asymmetric, and the antisymmetric component of the profile is determined by the dislocation polarization. It is shown that the strains in nanoparticles resulted by the relaxation of their surfaces also lead to line broadening.

Evaluation Methods of Line Profiles

The evaluation procedures of X-ray line profiles are overviewed in this chapter. These methods can be classified into four groups, namely (1) the most simple methods that evaluate only the breadths of diffraction peaks, (2) procedures using the Fourier-transforms of line profiles for the determination of the parameters of microstructures, (3) variance methods evaluating the restricted moments of peaks, and (4) procedures fitting the whole diffraction pattern. The crystallite size distribution and the densities of lattice defects cannot be determined from the peak width alone as the rule of summation of breadths of size, strain, and instrumental profiles depends on their shape. However, the breadth methods can be used for a qualitative assessment of the main origins of line broadening (size, dislocations, planar faults) (e.g. for checking the model of microstructure used in whole powder pattern fitting procedures). The application of Fourier and variance methods is limited if the diffraction peaks are overlapping. In the case of pattern fitting procedures, usually a microstructure model is needed for the calculation of the theoretical fitting functions. The reliability of these methods increases with increasing the number of fitted peaks.

Influence of Chemical Heterogeneities on Line Profiles

The chemical composition fluctuation in a material may cause line broadening due to the variation of the lattice parameter, which yields a distribution of the profile centers scattered from different volumes of the material. The nature of line broadening induced by chemical heterogeneities is similar to a microstrain-like broadening in the sense that the peak width increases with the magnitude of the diffraction vector. However, the dependence of compositional broadening on the orientation of diffraction vector (i.e. the anisotropic nature of this effect) differs very much from other types of strain broadening (e.g. from that caused by dislocations). The anisotropic line broadening caused by composition fluctuation is parameterized for different crystal systems and incorporated into the evaluation procedures of peak profiles. This chapter shows that the composition probability distribution function can be determined from the moments of the experimental line profiles using the Edgeworth series. The concentration fluctuations in decomposed solid solutions can also be determined from the intensity distribution in the splitted diffraction peaks.

Crystallite Size Broadening of Diffraction Line Profiles

In this chapter, the X-ray peak profile broadening caused by the finite size of scattering crystallites is studied in detail. According to Bertaut’s theorem, the line profile with the indices hkl is determined by the length distribution of columns building up the scattering crystallites normal to the hkl reflecting planes. The column length distribution determined from line profiles can be converted into crystallite size distribution. The effect of median and variance of crystallite size distribution on the shape of line profiles is also discussed. The line shapes for different crystallite size distribution functions (e.g. lognormal and York distributions) are given. It is shown that for spherical crystallites the peak broadening does not depend on the indices of reflections. The dependence of line profiles on the indices hkl is presented for various anisotropic shapes of crystallites.