Ground-state structures and the random-state energy of the Madelung lattice

We consider the classic Madelung problem of a lattice with N sites labeled i, each occupied by either an A or a B atom, and bearing a point charge ${\mathit{Q}}_{\mathit{i}}$ that depends on the environment of i. We find that, out of the $2^{\mathit{N}}$ possible lattice configurations of this binary ${\mathit{A}}_{1\mathrm{-}\mathit{x}}$ ${\mathit{B}}_{\mathit{x}}$ fcc alloy, the lowest-energy ‘‘ground-state structures’’ are the ${\mathit{A}}_3$B-, ${\mathit{A}}_2$ ${\mathit{B}}_2$- and ${\mathit{AB}}_3$-ordered superlattices with ordering vector (1,0,1/2). On the other hand, for the pseudobinary ${\mathit{A}}_{1\mathrm{-}\mathit{x}}$ ${\mathit{B}}_{\mathit{x}}$C zinc-blende alloy, the ground state corresponds to phase separation into AC+BC. Contrary to the accepted view, the Madelung energy of the random binary alloy is found to be nonvanishing.