Unequal wave vectors in short- versus long-range ordering in intermetallic compounds

The sign of the formation energy {Δ}HF of a compound indicates if the low-temperature long-range order (LRO) corresponds to compound formation (when {Δ}HF0) or to phase-separation (when {Δ}HF>0). However, {Δ}HF by itself does not tell us what type (ordering or clustering) of high-temperature short-range order (SRO) can be expected. The reason is that {Δ}HF contains two types of contributions: an ``elastic,'' volume-deformation energy G(x) that reflects the energy invested in deforming the constituents A and B to the volume V(x) of the A_1-x{B}ₓ$ alloy, and a ``chemical'' energy {ε} that reflects A-B interactions (charge transfer and atomic relaxations) at a fixed V(x). We show that the LRO and the SRO have the same behavior only when the signs of {Δ}HF and {ε} are the same: ordering tendencies (``type I'') when both {Δ}HF0 and {ε}0, or phase-separating/clustering tendencies (``type III'') when both {Δ}HF>0 and {ε}>0. However, ``type II'' systems ({Δ}HF>0; {ε}0) can exhibit a phase-separating LRO and an ordering-type SRO. Direct self-consistent local-density calculations of the total energy coupled with Monte-Carlo simulated annealing calculations of the ensuing Ising-like cluster expansion illustrate these type I, II, and III behaviors in Ni-V, Ni-Au, and Pd-Rh, respectively.