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.
TopGeneral Effect Of Lattice Distortions On Line Profiles
The atomic positions in a real crystal deviate from a perfect order as the external and/or internal stresses yield displacement of atoms from their ideal positions. These lattice distortions are usually caused by lattice defects, such as dislocations, but they may have other sources such as surface relaxation in nanoparticles. The displacement of lattice points from their ideal positions may cause both shift and broadening of the diffraction peaks. The first and the second effects are related to the change of the average spacing between atoms due to stresses and the variance in the interatomic spacing, respectively. In order to separate these two effects, we will introduce the concept of the ideal average lattice (Guinier, 1963). First, let us imagine an ideal crystal with perfect atomic order (see Figure 1a). Then, displace the lattice points (atoms) away from their theoretical positions as shown in Figure 1b. It is noted that in a real crystal the displacements are small compared to the interatomic spacings. An ideal average lattice can be constructed throughout the crystal for which the vectorial sum of the atomic displacements is zero, i.e. 
(Guinier, 1963). The ideal average lattice is illustrated by dashed lines in Figure 1c. Then, the diffraction peak shifts can be obtained by the difference in lattice spacings between the original and the average crystals, while the diffraction line broadening can be related to the displacements in the ideal average lattice. The strain causes peak shift or line broadening is usually referred to as macro- or microstrain, respectively. In the following, we will deal only with the description of the peak profile broadening caused by the lattice distortions.
Figure 1. Illustration of the construction of an ideal average lattice: a) distortion-free original crystal, b) displacements of lattice points relative to the distortion-free lattice, c) the ideal average lattice is shown by dashed lines for which the vectorial sum of the atomic displacements is zero, i.e. 
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