Quasiperiodic Frank-Kasper phases derived from the square-triangle dodecagonal tiling

Abstract

Frank-Kasper (F-K) phases form an important set of large-cell crystalline structures describing many inter-metallic alloys. They are usually described in term of their atomic environments, with atoms having $12, 14, 15$ and $16$ neighbours, coded into the canonical $Z_p$ cells (with $p$ the coordination number), the case $p=12$ corresponding to a local icosahedral environment. In addition, the long range structure is captured by the geometry of a network (called either "major skeleton" or "disclination network") connecting only the non-icosahedral sites (with $p\ne 12$). Another interesting description, valid for the so-called "layered F-K phases", amounts to give simple rules to decorate specific periodic 2d tilings made of triangles and squares and eventually get the 3d periodic F-K phases. Quasicrystalline phases can sometime be found in the vicinity, in the phase diagram, of the F-K crystalline alloys; it is therefore of interest to understand if and how the standard F-K construction rules can be generalized on top of an underlying quasiperiodic structure. It is in particular natural to investigate how well square-triangle quasiperiodic tilings with dodecagonal symmetry, made of square and (equilateral) triangles, can be used as building frames to generate some F-K-like quasicrystalline structures. We show here how to produce two types of such structures, which are quasiperiodic in a plane and periodic in the third direction, and containing (or not) $Z_{16}$ sites.