Strain coupling as the dominant interaction in structural phase transitions

A rigorous theory is developed for the ordering interaction J(R(ij)) in a crystal having a structural phase transition when J (R(ij)) is mediated by elastic relaxation in the material. The ordering process in cell i sets up a local stress field due to the sizes, shapes or displacements of atoms or atomic groups, which propagates elastically to a distant cell j. The atomistic theory for ferro- and antiferro-elastic transitions takes into account two types of singularity, one due to elastic anisotropy and the other to the Zener interaction J(Z) of infinite range in ferroelastic transitions, as well as the self-energy of relaxation around each cell. Four types of case are distinguished for a simple cubic model, which between them encompass the phenomena in much more complex situations. The interaction J(k) in Fourier space is dominated by whether or not domain walls perpendicular to k have a low energy from their strain satisfying Sapriel's compatibility relations. Thus embryonic tweed texture in fluctuations above T-c is readily accounted for. The asymptotic J(R) at large R is shown to be very anisotropic even in sign. The transition temperature T-c for ferro transitions in the mean-field (Bragg-Williams) approximation is dominated by the Zener contribution. The long-range and anisotropic nature of the coupling has implications for the kinetics of phase transformations, critical fluctuations near T-c, the theory of domain walls, and the formation of metastable textures, including 'tweed'.