Tile Hamiltonian for Decagonal AlCoCu

TL;DR

This paper develops a tile Hamiltonian model for decagonal Al-Co-Cu quasicrystals, capturing atomic interactions through effective tile interactions, and reveals how chemical ordering influences tile edge orientations and matching rules.

Contribution

It introduces a tile Hamiltonian for Al-Co-Cu quasicrystals based on ab-initio calculations, linking chemical ordering to tile edge interactions and quasiperiodic matching rules.

Findings

Dominant term counts H, B, S tiles.

Phason flips involving HS and BB pairs lower energy.

Edge orientation interactions enforce matching rules.

Abstract

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 {\em ab-initio} methods. The dominant term in the TH counts the number of H, B and S tiles. 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 tile edges. The edge arrow orientation in our model of AlCoCu is due to Co/Cu chemical ordering. Tile edges meet in vertices with 72$^\circ$ or 144$^\circ$ angles. We find strong interactions between edge orientations at 72$^\circ$ vertices that force a type of matching rule. Interactions at 144$^\circ$ vertices are somewhat weaker.