Higher-dimensional crystallography of N-fold quasiperiodic tilings.

Crystallography and periodic average structures (PASs) of two-dimensional (2D) quasiperiodic tilings with N-fold symmetry (N-QPTs with N = 7, 8, 9, 10, 11, 12, 13, 15) were studied using the higher-dimensional approach. By identifying the best (most representative) PASs for each case, it was found that the complexity of the PASs and the degree of average periodicity (DAP) strongly depend on the dimensionality and topology of the hypersurfaces (HSs) carrying the structural information. The distribution of deviations from periodicity is given by the HSs projected upon physical space. The 8-, 10- and 12-QPTs with their 2D HSs have the highest DAP. In the case of the 7-, 9-, 11-, 13- and 15-QPTs, the dimensionality of the HSs is greater than two, and is therefore reduced in the projection upon 2D physical space. This results in a non-homogeneous distribution of deviations from the periodic average lattice, and therefore in a higher complexity of the PASs. Contrary to the 7- and 9-QPTs, which still have representative PASs and DAPs, the 11-, 13- and 15-QPTs have a very low DAP.

Unifying cluster-based structure models of decagonal Al-Co-Ni, Al-Co-Cu and Al-Fe-Ni.

The geometrical building principles of Al-based decagonal quasicrystals and their approximants are discussed from a cluster-based approach. Our investigations cover 11 modifications with two- or four-layer periodicity in the systems Al-Co-Ni, Al-Co-Cu and Al-Fe-Ni. We identified a cluster that leads to a unifying view of all these phases. This unit cluster has ~20 Å diameter, four-layer periodicity along its tenfold axis and rod symmetry group p102m. The models obtained are in agreement with all the electron-density maps and electron-microscopy images available.

Colloidal quasicrystals with 12-fold and 18-fold diffraction symmetry.

Micelles are the simplest example of self-assembly found in nature. As many other colloids, they can self-assemble in aqueous solution to form ordered periodic structures. These structures so far all exhibited classical crystallographic symmetries. Here we report that micelles in solution can self-assemble into quasicrystalline phases. We observe phases with 12-fold and 18-fold diffraction symmetry. Colloidal water-based quasicrystals are physically and chemically very simple systems. Macroscopic monodomain samples of centimeter dimension can be easily prepared. Phase transitions between the fcc phase and the two quasicrystalline phases can be easily followed in situ by time-resolved diffraction experiments. The discovery of quasicrystalline colloidal solutions advances the theoretical understanding of quasicrystals considerably, as for these systems the stability of quasicrystalline states has been theoretically predicted for the concentration and temperature range, where they are experimentally observed. Also for the use of quasicrystals in advanced materials this discovery is of particular importance, as it opens the route to quasicrystalline photonic band gap materials via established water-based colloidal self-assembly techniques.

Fascinating quasicrystals.

It took Dan Shechtman more than two years to get his discovery of an Al-Mn phase with icosahedral diffraction symmetry and sharp Bragg reflections published. A paradigm shift had to take place before this novel ordering state of matter - seemingly contradicting crystallographic laws - could be accepted. Today, more than 25 years later, the existence of quasicrystals is beyond doubt. However, not everything is settled yet. All the factors governing formation, growth, stability and structure of quasicrystals are still not fully understood, nor is it resolved whether their structures are strictly or only on average quasiperiodic, and it is still an open question why only quasicrystals with 5-, 8-, 10- and 12-fold rotational symmetry have been experimentally observed so far. These points will be addressed in this review article.