A Theory of Anharmonic Lattice Statics for Analysis of Defective Crystals

In this topic we present a general theory of anharmonic lattice statics for analysis of defective complex lattices. This theory differs from the classical treatments of lattice statics in that it does not rely on knowledge of force constants for a limited number of nearest neighbor interactions. Instead, the only thing needed as input is an interatomic potential that models the interaction of atoms this theory takes into account the fact that close to defects force constants are different from those in the bulk crystal. This formulation of lattice statics reduces the analysis of defective crystals to solving discrete boundary-value problems which consist of system of difference equations with some boundary conditions. To be able to solve the governing equations analytically, the discrete governing equations are linearized about a reference configuration that resembles a nominal defect. Fully nonlinear solutions are obtained by modified Newton-Raphson iterations of the harmonic solutions. In this theory, defective crystals are classified into three groups: defective crystals with 1-D symmetry reduction, defective crystals with 2-D symmetry reduction, and defective crystals with no symmetry reduction. Our theory systematically reduces the discrete governing equations for defective crystals with 1-D and 2-D symmetry reductions to ordinary difference equations and partial difference equations in two independent variables, respectively. Solution techniques for the discrete governing equations are demonstrated through some examples for ferroelectric domain walls. This formulation of lattice statics is very similar to continuum mechanics and we hope that developing this theory would be one step forward for doing lattice scale calculations analytically