Analysis and modelling of structural disorder by the use of the three-dimensional pair distribution function method exemplified by the disordered twofold superstructure of decagonal Al–Cu–Co

Theoretical concepts and the practical application of the three-dimensional pair distribution function (3D-PDF) method and its variant, the three-dimensional Δ-pair distribution function (3D-ΔPDF) method, are presented. In analogy to traditional Patterson function analysis, advantage is taken of the Fourier transformation either of the full three-dimensional diffraction pattern of a disordered crystal or just of the isolated diffuse scattering, respectively. By the use of three-dimensional information, analysis of disorder becomes straightforward, and it becomes possible to investigate far more complicated structures than is feasible with well established powder diffraction-based PDF analysis. Compared to more traditional modelling techniques, such as Monte Carlo simulation, the 3D-ΔPDF provides direct access to disorder models and allows selective modelling of distinct structural features, which are, in contrast to reciprocal space, well localized in PDF space. The principles of the 3D-ΔPDF approach are exemplified using an analysis of the twofold (∼8 Å) periodic superstructure of a decagonal Al₆₅Cu₂₀Co₁₅ quasicrystal. Although analysis of disorder in quasicrystals is far more demanding than in the case of periodic structures, details of the disordered structure could be elucidated. The superstructure is found to be built from columnar units, having a maximum diameter of ∼14.5 Å. The lateral correlation between these columns is weak. Internally, the columns consist of a long-range-ordered alternation of flat and puckered layers. The development of the model and the atomic structure of the columns are described in detail.