Tile Hamiltonian for decagonal AlCoCu derived from first principles

A tile Hamiltonian (TH) replaces the actual atomic interactions in a quasicrystal with effective interactions, between and within tiles. We studied Al-Co-Cu decagonal quasicrystals described as decorated hexagon-boat-star (HBS) tiles using ab initio methods. The dominant term in the TH counts the number of H, B, and S tiles. The phason flips that replace an HS pair with a BB pair lower the energy. In Penrose tilings, quasiperiodicity is forced by arrow matching rules on the tile edges. The edge arrow orientation in our model of AlCoCu is due to Co/Cu chemical ordering. The tile edges meet in vertices with 72^∘ or 144^∘ angles. We find strong interactions between edge orientations at 72^∘ vertices that force a type of matching rule. Interactions at 144^∘ vertices are somewhat weaker.