Relaxations near surfaces and interfaces for first-, second- and third-neighbour interactions: theory and applications to polytypism

Structural relaxations near surfaces and interfaces are analysed in a simple, generic model with first-, second- and third-layer interactions. The relaxations have exponential envelopes with three types of structural distortion: (i) ferrodistortive, (ii) antiferrodistortive, and (iii) modulated (incommensurable). Their stability conditions in the field of control parameters and their relationship with structural phase transitions is derived. A tricritical point is found for three-layer interactions and a transition to an incommensurate phase. All phase transitions to the ferrodistortive phase are first-order in the model. The theory is applied to the analysis of lattice relations near internal interfaces in ionic polytypic materials (e.g. PbI2). Diffuse X-ray scattering and the shift of diffraction angles are the typical fingerprints for such relaxations. The relevant structure factors are calculated. There is tentative agreement between the calculated and observed diffraction profiles.