Ab initio study of long-period superstructures in close-packed ${\mathit{A}}_3$B compounds

We have performed ab initio calculations of the stability of one-dimensional long-period superstructures in ${\mathrm{Cu}}_3$Pd, ${\mathrm{Cu}}_3$Al, and ${\mathrm{Ag}}_3$Mg by means of an interface Green’s function technique based on the linear-muffin-tin-orbitals method within the tight-binding and atomic-sphere approximations. The energies of the superstructures relative to the L$1_2$ structure are found by an expansion based on the calculated energy of a single (001) antiphase boundary and the calculated interaction energy between two and three antiphase boundaries of varying distance. The expansion agrees with standard band-structure calculations of the structural energy differences for the two short-period superstructures ${\mathit{DO}}_{22}$ and ${\mathit{DO}}_{23}$. We find that at zero temperature the ground states of ${\mathrm{Cu}}_3$Pd, ${\mathrm{Cu}}_3$Al, and ${\mathrm{Ag}}_3$Mg are one-dimensional superstructures with antiphase boundary separations of 2–5 unit cells of the underlying L$1_2$ structure.