First Principles Phase Diagram Calculations for the Octahedral-Interstitial System $Hf{O}_X$, $0\le X\le 1\mathrm{/}2$

First principles based phase diagram calculations were performed for the hexagonal closest packed octahedral-interstitial solid solution system $\alpha HfO_{X} ($\alpha Hf[{]}_{1-X}{O}_X$; [ ]=Vacancy; $0\le X\le 1\mathrm{/}2$). The cluster expansion method was used to do a ground state analysis, and to calculate the phase diagram. The predicted diagram has four ordered ground-states in the range $0\le X\le 1\mathrm{/}2$, but one of these, at X=5/12, is predicted to disproportionate at T$\approx 220K$. At X$\approx 1\mathrm{/}3($Hf_{3}O$) and X$\approx 1\mathrm{/}2($Hf_{2}O$), order-parameter vs temperature plots predict Devil’s Staircases of closely related ordered structures.