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 Qi that depends on the environment of i. We find that, out of the 2N possible lattice configurations of this binary A1-xBx fcc alloy, the lowest-energy ''ground-state structures'' are the A3B-, A2B2- and AB3-ordered superlattices with ordering vector (1,0,1/2). On the other hand, for the pseudobinary A1-xBxC 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