Grid-increment cluster expansion for polymorphic structures in alloys

We propose a cluster expansion (CE) technique that can be applied to any atomic arrangement on alloy systems in their polymorphs. The proposed CE introduces virtual vacancy sites in the starting structure in order to describe the atomic positions for a variety of lattices. We derive the general condition for obtaining effective interactions in the present CE, which results in rigorous expansion of any properties on polymorphs in alloys. As an example, the proposed CE is combined with a density functional theory (DFT) calculation, applied to the description of internal energy for the Al–Na binary alloy system, which clarifies the concept and practical application of the CE to real systems. We successfully predict the tendency of phase separation as well as the most stable structures for Al (fcc) and Na (bcc) from information about the internal energy for structures other than fcc or bcc.