This chapter analyzes the thermodynamic stability of “static concentration waves.” The idea is that an ordered structure can be described as a variation of chemical composition from site to site on a crystal lattice, and this variation can be written as a wave, with crests denoting B-atoms and troughs the A-atoms, for example. The wave does not propagate, so it is called a “static” concentration wave. Another important difference from conventional waves is that the atom sites are exactly on the tops of crests or at the bottoms of troughs, so we do not consider the intermediate phases of the concentration wave, at least not in our main examples. A convenient feature of this approach is that an ordered structure can be described by a single wavevector, or a small set of wavevectors. The disordered solid solution has no such periodicity, so the amplitude of the concentration wave, η, serves as a long-range order parameter.

This chapter begins with a review of how periodic structures in real space are described by wavevectors in k-space, and then explains the “star” of the wavevector of an ordered structure. A key step for phase transitions is writing the free energy in terms of the amplitudes of static concentration waves.