When minerals become complex: an elementary introduction to superspace crystallography to describe natural-occurring incommensurately modulated structures

Aperiodic crystals are compounds with perfect long-range order of their atomic structures, but without three-dimensional translational symmetry. In the last thirty years, incommensurate crystal structures have been discovered in many compounds from synthetic inorganic and organic compounds to proteins. With the introduction of the superspace theory it was realized that any crystal structure requiring more than three integers to index its diffraction pattern can be described as a periodic object in superspace with dimension equal to the number of the required integers. The structure observed in physical space is a three-dimensional intersection of the structure described as periodic in superspace. In recent years, the use of CCD and imaging plate systems considerably changed the sensitivity of data collection for modulated structures and, therefore, there was a need for further improvement of the methods. Today, several computer programs are able to solve and refine incommensurately modulated structures using the superspace approach. In nature, it is uncommon to find minerals having strong and sharp incommensurate satellites that could be used for a multidimensional refinement. As a very clear example, the case of natural akermanite, Ca(2)MgSi(2)O(7), which belongs to the melilite-group of minerals, will be shown in detail. The five-dimensional refinement carried out on this crystal shows that the displacive modulation of the atoms is mainly related to a variation of the Ca cation coordination.