The continuous melting transition of a three-dimensional crystal at a planar elastic instability

The structure and scattering cross section are analysed for a crystal undergoing an elastic instability in which the elastic constant decreases to zero over a plane or planes of propagation directions. It is shown that, at this phase transition, long-range order is destroyed and that the crystal undergoes a novel form of melting. In the scattering cross section, the Bragg peaks disappear and are replaced by power-law singularities which are analogous to those occurring in the scattering from a smectic liquid crystal or a two-dimensional crystal. Expressions are obtained to describe the behaviour as the phase transition is approached.