Powder diffraction characterization of stacking disorder

Determination of stacking ordering in disordered close-packed structures from pairwise correlation functions

It is shown how to reconstruct the stacking sequence from the pairwise correlation functions between layers in close-packed structures. First, of theoretical interest, the analytical formulation and solution of the problem are presented when the exact pairwise correlation counts are known. In the second part, the practical problem is approached. A simulated annealing procedure is developed to solve the problem using as initial guess approximate solutions from previous treatments. The robustness of the procedure is tested with synthetic data, followed by an experimental example. The developed approach performs robustly over different synthetic and experimental data, comparing favorably with the reported methods.

A Markov theoretic description of stacking-disordered aperiodic crystals including ice and opaline silica

This article reviews the Markov theoretic description of one-dimensional aperiodic crystals, describing the stacking-faulted crystal polytype as a special case of an aperiodic crystal. Under this description the centrosymmetric unit cell underlying a topologically centrosymmetric crystal is generalized to a reversible Markov chain underlying a reversible aperiodic crystal. It is shown that for the close-packed structure almost all stackings are irreversible when the interaction reichweite s > 4. Moreover, the article presents an analytic expression of the scattering cross section of a large class of stacking-disordered aperiodic crystals, lacking translational symmetry of their layers, including ice and opaline silica (opal CT). The observed stackings and their underlying reichweite are then related to the physics of various nucleation and growth processes of disordered ice. The article discusses how the derived expressions of scattering cross sections could significantly improve implementations of Rietveld's refinement scheme and compares this Q-space approach with the pair-distribution function analysis of stacking-disordered materials.

Extrinsic faulting in 3C close-packed crystal structures: computational mechanics analysis

Extrinsic faulting has been discussed previously within the so-called difference method and random walk calculation. In this contribution it is revisited under the framework of computational mechanics, which allows expressions to be derived for the statistical complexity, entropy density and excess entropy as a function of faulting probability. The approach allows one to compare the disordering process of an extrinsic fault with other faulting types. The ℇ-machine description of the faulting mechanics is presented. Several useful analytical expressions such as probability of consecutive symbols in the Hägg coding are presented, as well as hexagonality. The analytical expression for the pairwise correlation function of the layers is derived and compared with results previously reported. The effect of faulting on the interference function is discussed in relation to the diffraction pattern.

The “state of the art” of the diffraction analysis of crystallite size and lattice strain

This paper addresses both old, but “renovated” methods and new methods for diffraction line-profile analysis. Classical and even extremely simple single-line methods for separating “size” and “strain” broadening effects have merit for characterization of the material imperfectness, but it is generally very difficult to interpret the data obtained in terms of microstructure parameters as used in materials science. Developments of recent years, focusing on distinct anisotropic line-broadening effects, as due to the type, orientation and distribution of dislocations and minute compositional variation, will be touched upon. The most promising development may be the synthesis of line profiles on the basis of a microstructure model and application of the (kinematical) diffraction theory without any further assumption, which contrasts with the other methods. This approach can in principle be applied in single-line and multiple-line variants and also in analyses of the whole diffraction pattern. The advantage is the direct evaluation of microstructure parameters as used in materials science. The challenge is to develop microstructure models which are flexible enough to be applicable in more than one case ...