Modern crystallography faces a demanding challenge of describing atomic structure and diffraction pattern of quasicrystals, which, after 30 years of Shechtman’s discovery, is still an open field of research. The classical approach based on the Braggs and Laue equations in three-dimensional space is useless, because the direct and the reciprocal lattices cannot be introduced for aperiodic systems. A standard solution to this problem, applied by number of scientists, is to retrieve periodicity in high dimensions. This is a purely mathematical approach with some difficulties from a point of view of physics. It is mathematically elegant, but not applicable to all aperiodic systems (e.g. Thue-Morse or Rudin-Shapiro sequences). It meets also a serious trouble in a proper description of structural defects, like phasons. In our opinion the most successful alternative to the multidimensional description is a statistical method of diffractional and structural analysis of aperiodic systems, also known as the average unit cell approach (AUC). In this work an application of the AUC method to selected aperiodic systems, including modulated structures, quasicrystals and covering clusters, is discussed in the form of a mini-review. A reader can find more details in the cited references.

One of the challenges of modern crystallography of complex systems (complex metallic alloys, proteins, aperiodic crystals and quasicrystals) is to properly describe the disorder in these systems and discuss correctly the refinement results in terms of the structural disorder. In this paper we briefly discuss a new approach to phasons and phonons in quasicrystals and focus on the new theory of phonons in these materials. A newly derived correction factor for phonons in the form of the Bessel function is the approximated way of describing optic modes in the phonon spectra of quasicrystals. It is applied to a real decagonal quasicrystal in the Al-Cu-Rh system with 56/38 atoms per thick/thin structural unit, based on 2092 unique reflections selected from the collected diffraction data, significantly improving the refinement results. The final R-factor value is 7.24%, which is over 0.5% better result comparing to originally reported. We believe our work will open a broader discussion on the disorder in quasicrystals (and other aperiodic systems) and motivate to develop new approaches to treat the diffraction data influenced by different types of disorder in the new way.

The effect of replacing iron with transition metals (M = Mn, Cr, Co) on the microstructure of mechanically alloyed Al65Cu20Fe15 quasicrystalline powder was examined by X-ray diffraction and transmission electron microscopy methods. Powders of various compositions were milled in a high-energy planetary ball mill up to 30 hours at a rotation speed 350 rpm using WC milling media. The amount of the fourth additions was constant in all powders and Fe atoms were replaced with Mn, Cr or Co in a 1:1 ratio, while the content of the Al and Cu was selected in two ways: they remained the same as in the initial ternary Al65Cu20Fe15 alloy or changed to obtain e/a ratio = 1.75 (optimal for icosahedral quasicrystalline phase). Quasicrystalline phase formed in the quaternary Al65Cu20Fe7.5M7.5 powders, whereas in the second group of compositions only crystalline phases were identified.