The interactions between the lattice planes in planar lamellar crystalline systems in which the interatomic potentials are of electrostatic origin or in which they result from a coupling with elastic deformations show a distance dependence given by a series of exponentially decreasing functions. In the vicinity of the two-dimensional (2D) Brillouin zone and near the points of elastic instabilities the exponential decrease of some of the terms of the series is fairly slow. Two methods are presented which allow us to obtain closed expressions for the Green functions in such systems with and without surfaces regardless of the rate of the exponential decay. The first method, useful for exclusively exponential interactions, reduces the problem to the multiplication of matrices known from the theory for short-range interactions. The second method is adapted to systems having short- and long-range interactions at the same time and consists in introducing some additional degrees of freedom by which the range of the interactions becomes finite. Profiles of the order parameter at domain walls and surface relaxation in a crystal with short- and long-range interactions are calculated as first applications of the methods. A crossover between ferrodistortive and antiferrodistortive responses and anomalous localization of the response is found in systems with competing ferrodistortive and antiferrodistortive tendencies between the elasticity and the rotation-translation coupling. The transformation enabling the introduction of the additional degrees of freedom in the case of dipolar interactions is given.

• Received 1 February 1993

DOI:https://doi.org/10.1103/PhysRevB.47.16447