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 intermetallic 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 whether 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.