Ivan Orlov
The superspace concept is a new crystallographic approach for the generalisation of symmetry, structural models and structure dependent properties. Although the original motivation for the superspace was intended to describe symmetry of aperiodic crystals, the concept gained general acceptance in a wider range of applications. In particular, the method is able to combine structures into a single model, thus offering a clearer view of their hidden relationships. While the individual members of the family can have different space group symmetry and be both periodic or aperiodic, the generalization in superspace leads to a common crystallographic structure model and a common superspace group reuniting all of them. The benefits of such model are best demonstrated, if applied to an extensive and diverse family of compounds. Hexagonal ferrites is such an extensive family and a prominent example. Ferrites are widely known for their applications in motors, consumer electronics, microwave technology and, recently, in medicine. In general ferrites are complex oxides composed of various metals and oxygen. "Hexagonal" ferrites, named in opposition to cubic or spinel ferrites, form a particular group closely related to the mineral magnetoplumbite with the approximate chemical composition PbFe12O19. Their high uniaxial magnetocrystalline anisotropy renders them particularly useful for electronics. Currently the family includes more than 60 members of magnetic materials with unit cell dimension extending up to 1577 Å. The study of compounds approaching such a "biological magnitude" is necessarily complicated by different symmetries of individual structures and by the large number of possible structure models, generated by different stacking of several basic units. Fortunately, the generalization in higher dimensional space offers a mechanism for solving these difficulties. The description of the family is not only more elegant but also reveals characteristic relations which are not easily observable while dealing with individual structures. Understanding the "code" of such features permits a direct readout of crystal structures and unique structure solutions for this realm of inorganic species. In this work, the structures of the hexagonal ferrites are treated in superspace as a structural modulation of a common underlying average structure. The average layer-to-layer distance is directly related to the average structure periodicity along the layer stacking direction. In a first approximation series of equidistant layers are stacked along a specific direction. A modulation is introduced by varying the type of each layer. The fundamental atomic modulation is therefore of occupational nature and can be described by means of step-like functions, which define discontinuous atomic domains in superspace. The higher-dimensional description leads to the diffraction pattern indexed with four indices hklm. However, this indexing is not unique, providing room for various models. An appropriate choice of the model is based on common structural features of the described compounds and their embedding in superspace. In order to simplify the search for a common superspace group for a set of given space groups, especially for future superspace embeddings, a database providing analytical relations between (3+1)-dimensional superspace groups and three-dimensional space groups has been created and is now available on the World Wide Web. The first complete derivation of a subgroup-supergroup tree for (3+1)-dimensional symmetry made in the course of this thesis provided an informational ground for such database. The results obtained in the present thesis demonstrate that the superspace approach is an appropriate and powerful tool for analysing compounds sharing common structural features. Both periodic and aperiodic ferrite structures, which were investigated in the course of this work, can be better understood by the description in (3+1)-dimensional superspace.
EPFL2007
