Electron diffraction of commensurately and in- commensurately modulated materials. Joke Hadermann and Artem M. Abakumov EMAT, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp 1. 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 periodici- ty 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 crys- tals 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 re- ciprocal lattice using the ED data is a great help for solving these problems, espe- cially taking into account that powder diffraction data often suffer from poor reso- lution (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 struc- ture, such as displacements of the oxygen atoms or oxygen/vacancy ordering. These lecture notes are 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. 2. Commensurately modulated structures 2.1. The supercell approach Assume a simple primitive structure I with only one atom type A, as shown in Fig.1a. For simplicity we will consider a two-dimensional lattice, but the consid- eration can be extended to the third axis analogously. The schematic representa- tion of the ED pattern of this structure, is given next to the scheme of structure I