Identification of coherent twin relationship from high-resolution reciprocal space maps

Abstract

Twinning is a common crystallographic phenomenon, which usually occurs in crystals during symmetry-lowering phase transition. Once formed, twin domains play an important role in defining physical properties: for example, twin domains underpin the giant piezoelectric effect in ferroelectrics, superelasticity in ferroelastics and the shape-memory effect in martensitic alloys. Unfortunately, there is still a lack of experimental methods for imaging and characterization of twin domain patterns. Here, we propose a theoretical framework and an algorithm for the recognition of twinned pairs of ferroelastic domains and the identification of the coherent twin relationship using high-resolution reciprocal space mapping of X-ray diffraction intensity around split Bragg peaks. Specifically, we adapt the geometrical theory of twinned ferroelastic crystals (Fousek & Janovec, 1969) for the analysis of the X-ray diffraction patterns. We derive the necessary equations and outline an algorithm for calculation of the separation between the Bragg peaks, diffracted from possible coherent twin domains, connected to one another via mismatch-free interface. We demonstrate that such separation is always perpendicular to the planar interface between mechanically matched domains. As examples, we present the analysis of the separation between the peaks diffracted from tetragonal and rhombohedral domains in the high-resolution reciprocal space maps of BaTiO3 and PbZr1-xTixO3 crystals. The demonstrated method can be used to analyse the response of multi-domain patterns to external perturbations such as electric field, change of a temperature or pressure.