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.
TopSize Broadening Of A Small Crystal
According to the kinematical theory of X-ray diffraction, the amplitude of the elastically scattered X-rays (Ψ) is proportional to the Fourier-transform of the electron density, η(r) (Warren, 1990):
, (2.1) where F denotes the Fourier transformation, r and κ are arbitrary vectors in the crystal and the reciprocal spaces, respectively, and the integration is carried out for the whole volume of the crytallite. η(r) is defined as the number of electrons in a unit volume at position r. The electron density is a periodic function of r, since a crystallite is a periodic structure of cells. Therefore, the electron density at r can be expressed as a sum of the electron densities originating from the different cells in the crystallite (see Figure 1):
, (2.2)
Figure 1. Schematic picture illustrating that the electron density at position r can be expressed as the sum of electron densities in the different cells of the crystallite
where
ηc(
r) is the electron density in a cell and
Rn is a lattice space vector representing the position of the
nth cell (
n is the index of the cell).