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

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 r a , r b , r c are the principal ellipsoid radii, and c a , c b , c c direction cosines of diffraction vector related to the principal axes of ellipsoid, the average dimension of crystallites along the diffraction vector ( D hkl ) is: D hkl = K / c a 2 / r a 2 + c b 2 / r b 2 + c c 2 / r c 2 . The coefficient K has the value 3/2 if D hkl 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: b 11 h 2 + b 22 k 2 + b 33 l 2 +2 b 12 hk +2 b 13 hl +2 b 23 kl = K 2 / L hkl 2 d hkl 2 , where b ij 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.