Use of an ellipsoid model for the determination of average crystallite shape and size in polycrystalline samples

Abstract

A mathematical model for interpreting the anisotropical broadening of the powder diffraction lines by an average crystallite in the form of a triaxial ellipsoid is developed. The model covers satisfactorily a broad range of averaged crystallite shapes in polycrystalline samples of all crystal symmetries and provides simple formulas for use in powder pattern fitting routines. When ra, rb, rc are the principal ellipsoid radii, and ca, cb, cc direction cosines of diffraction vector related to the principal axes of ellipsoid, the average dimension of crystallites along the diffraction vector (Dhkl) is: Dhkl=K/ca2/ra2+cb2/rb2+cc2/rc2. The coefficient K has the value 3/2 if Dhkl is the volume average dimension of crystallites along the diffraction vector, or 4/3 in the case of the surface average dimension. The appropriate expression for use in whole pattern fitting routines is: b11h2+b22k2+b33l2+2b12hk+2b13hl+2b23kl=K2/Lhkl2dhkl2, where bij are the elements of a second-rank symmetric tensor. Finding eigenvalues and vectors of tensor b gives dimensions and orientations of the principal ellipsoid radii in reciprocal lattice values.