Acoustic phonon instabilities and structural phase transitions

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Abstract
Structural phase transitions are considered in which the order parameter is a homogeneous deformation of the crystal. The fluctuations at these transitions are the acoustic modes, and it is shown that an effective Hamiltonian may be constructed describing the homogeneous deformations and their fluctuations. There are three cases which result, those in which there are no fluctuations with wavelengths less than the crystal dimensions, those in which the acoustic modes have strongly temperature-dependent velocities for wave vectors on particular lines of reciprocal space, and those for which the velocities are temperature dependent for wave vectors within planes in reciprocal space. In many cases, the transitions are expected to be first order because of the presence of cubic invariants in the effective Hamiltonian. In those which may be continuous, the behavior is shown by use of renormalization-group theory to be that of classical Landau theory, with the possibility of logarithmic corrections in one particular instance. Unfortunately, we are unaware of any examples of this case, but in the other cases, the results are in accord with experimental results.