A four-dimensional model for the Ba–Ti–O dodeca­gonal quasicrystal

A four-dimensional (4D) model is presented of the first oxide dodecagonal quasicrystal found in a Ba–Ti–O ultra-thin film on a Pt(111) single-crystal substrate. The 4D model, with a 4D dodecagonal lattice constant a ( d ) = 8.39 Å, was derived by considering a tile decoration model of dodecagonal Niizeki–Gähler tiling composed of squares, triangles and 30° rhombuses. The model consists of four kinds of occupation domain, and 4D positional vectors defining the shape of each occupation domain are given. Moreover, the atomic arrange­ment of two Ba–Ti–O periodic approximants, the sigma-phase approxi­mant and a 25.6 Å approximant were derived from the 4D model by the introduction of linear phason strains.