Electron Diffraction of Commensurately and Incommensurately Modulated Materials

Abstract

Many materials have structures which are modulated, either displacively or compositionally, or a combination of both. A distinction can be made between commensurately modulated structures (where the period of the modulation is an integral multiple of the periodicity of the unmodulated cell) and incommensurately modulated structures (where that period is a non-integral multiple of the periodicity of the underlying unmodulated cell).

Electron diffraction (ED) is often very helpful in the analysis of modulated structures, especially if the compound is not available in the form of single crystals suitable for X-ray diffraction. Being applied to modulated structures, powder diffraction techniques face problems in the correct determination of the periodicity of the modulation waves (reflected in the modulation vector(s) in the reciprocal space) and the symmetry associated with the basic structure and the modulation waves (reflected in the superspace symmetry group). Reconstruction of the 3D reciprocal lattice using the ED data is a great help for solving these problems, especially taking into account that powder diffraction data often suffer from poor resolution (neutron powder diffraction) or low signal/noise ratio (conventional laboratory X-ray diffraction). This creates particular difficulties in recognizing weak satellite reflections, caused by some minor perturbations of the basic structure, such as displacements of the oxygen atoms or oxygen/vacancy ordering.

This chapter is an introduction to handling electron diffraction patterns of both commensurately and incommensurately modulated materials. This does not extend to the underlying theory of incommensurate crystallography due to the limited amount of space, and appropriate references are given instead.