An exactly soluble model for distortive structural phase transitions in a crystal with a single defect

An exactly soluble three-dimensional spherical-like model is considered to describe the distortive structural phase transition of a crystal with a single defect. The defect can be incorporated in the vibrations of all other particles only if the attractive single-site potential of the defect is less than a certain critical value. No local distortions can appear above the phase transition temperature TC of the perfect host crystal. Above the critical value of its attractive single-site potential, the defect does not participate in the motions of the host. But a local mode may appear above TC which only condenses if the critical value is reached. The local mode can only soften at the phase transition temperature of the perfect system. All critical exponents of the model with a single defect are the same as the critical exponents of the perfect lattice.