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The phase separation behaviors from the single B2 ordered phase into the twoseparate B2 and L2$_1$ ordered phases in the X-Al-Ti (X: Fe, Co, and Ni) alloysare analyzed using the cluster variation method (CVM) with the interactionenergies evaluated from the electronic band structure calculations. The cubicapproximation of the CVM is employed for the X$_2$Al$_{2-x}$Ti$_x$ ($0 \leq x\leq 2$) alloys limiting an interchange between Al and Ti atoms on the$\alpha$- and $\beta$-sublattices of the L2$_1$ ordered structure with the Xatoms fixed on the $\gamma$-sublattice. The phase stabilities of the B2 andL2$_1$ structures are examined, and the phase diagrams at the pseudo-binarysection, XAl-XTi, are determined. The two-phase regions of B2 and L2$_1$phases, i.e., phase separation behavior, are successfully produced in both Co-and Ni-Al-Ti alloy systems, whereas no phase separation is predicted in theFe-Al-Ti alloy. The origin of the phase separation in the Co- and Ni-Al-Tialloys is, respectively, attributed to the mechanical instability and thecombination of mechanical instability and chemical repulsions of unlike pairs.

Origin of the Phase Separation into B2 and L21 Ordered Phases in the  X-Al-Ti (X: Fe, Co, and Ni) Alloys from the First-principles Cluster  Variation Method