Anisotropic diffraction-line broadening due to microstrain distribution: parametrization opportunities

A correlated Gaussian lattice-parameter distribution of an ensemble of crystals, as leading to line broadening in the course of powder diffraction, can be associated with a correlated Gaussian microstrain distribution. The latter can be described in terms of a fourth-rank covariance tensor containing as its 81 components E_ijpq, the variances and the covariances of the nine components [epsilon]_ij of the symmetric second-rank strain tensor (formulated with respect to Cartesian coordinates), i.e. E_ijpq = 〈[epsilon]_ij[epsilon]_pq〉. The restrictions for the E_ijpq tensor components resulting from assumed crystal class-symmetry invariance are the same as expected for certain fourth-rank property tensors, like compliancy. The parametrization of anisotropic microstrain broadening (e.g. in the course of Rietveld refinement) on the basis of the covariance tensor components E_ijpq has, in comparison with earlier approaches, the advantage of straightforward recognizability of the case of isotropic microstrain broadening, independently of the actual crystallographic coordinate system.