Order-disorder transition in the Cd-Ca cubic approximant

Recent experiments discovered an order-disorder transition occuring at low temperatures in large unit 1/1 cell cubic approximants of the stable Cd-based binary alloy quasicrystals. The transition is related to correlations among orientational degrees of freedom whose separations are around 12 \AA. We analyze the interactions between the degrees of freedom using {\em ab-initio} calculations for Cd-Ca alloys and derive an equivalent antiferromagnetic Ising model which shows a similar phase transition. However, the calculated transition temperature is higher than observed experimentally, indicating that the actual structure and its order-disorder transition are more complex than originally proposed. A side-benefit of our study is the discovery of a canonical-cell decoration model for the Cd-Ca icosahedral phase.