Polytypism in decagonal quasi-crystals and approximant crystals

A classification scheme, which is able to relate many crystalline compounds to the decagonal quasi-crystalline phases, is given based on a description of the atomic decoration in terms of a small number of subtiling units. This scheme is suitable for one-dimensional sequences, two-dimensional tilings and three-dimensional networks. The relative numbers of each type of tiling unit in the Penrose pattern are calculated. The paper first focuses on the two-dimensional aperiodic plane. In a way reminiscent of polytypism, a distinction is then made according to the way such planes are stacked periodically on top of each other, thus giving insight into the structure of the known decagonal phases. Emphasis is then put on the aperiodic scheme, which may be related to non-ideally ordered icosahedral phases. This way of describing the structure allows one to construct atomic models in three dimensions and to calculate accordingly the composition of the model structure starting from a known crystalline phase with identical or closely related composition. Many approximant and quasi-crystalline structures, not all yet observed experimentally, may be sorted out.