Crystallographic excursion in superspace

After studying more than 100 different samples of calaverite Au1-pAgpTe2 (p 2CO3 resisted any attempt for a precise structural analysis. The appearance of satellite reflections was noted on single crystal diffractograms which led to the generalisation of the concept of crystal. This generalisation consisted in using at least four integers to fully characterise each individual diffractions peaks. The theory of periodic crystals in space of higher dimension, i.e. the superspace was then developed to deal with the new experimental observations. Later, a new class of materials called composite crystals and still later, the discovery of quasicrystals only reinforced the validity of the superspace concept to describe any material requiring more than three integers to index their diffraction pattern.What is the essence of superspace to describe crystalline structures? Any crystal structure requesting 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 required integer. The structure observed in our real word is a three dimensional cut of the superspace description. In general this cut is irrational and consequently the crystal is aperiodic. Calaverite and [gamma]-Na2CO3 are examples of aperiodic crystals which includes incommensurately modulated crystals, composite and quasi-crystals. Rational cuts are also possible. In this case, the structure is periodic and is usually called a superstructure.

Application of the superspace approach to members of the ferrite family

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

Differential geometry applied to crystallography

Philippe Kocian

The mathematical facet of modern crystallography is essentially based on analytical geometry, linear algebra as well as group theory. This study endeavours to approach the geometry and symmetry of crystals using the tools furnished by differential geometry and the theory of Lie groups. These two branches of mathematics being little known to crystallographers, the pertinent definitions such as differentiable manifold, tangent space or metric tensor or even isometries on a manifold together with some important results are given first. The example of euclidean space, taken as riemannian manifold, is treated, in order to show that the affine aspect of this space is not at all an axiom but the consequence of the euclidean nature of the manifold. Attention is then directed to a particular subgroup of the group of euclidean isometries, namely that of translations. This has the property of a Lie group and it turns out that the action of its elements, as well as those of its Lie algebra, plays an important role in generating a lattice on a manifold and in its tangent space, too. In particular, it is pointed out that one and only one finite and free module of the Lie algebra of the group of translations can generate both, modulated and non-modulated lattices. This last classification therefore appears continuous rather than black and white and is entirely determined by the parametrisation considered. Since a lattice in a tangent space has the properties of a vector space, it always possesses the structure of a finite, free module, which shows that the assignment of aperiodicity to modulated structures is quite subjective, even unmotivated. Thanks to the concept of representation of a lattice or a crystal in a tangent space, novel definitions of the notions of symmetry operation of a space group and point symmetry operation, as well as symmetry element and intrinsic translation arise; they altogether naturally blend into the framework of differential geometry. In order to conveniently pass from one representation of a crystal in one tangent space to another or to the structure on a manifold, an equivalence relation on the tangent bundle of the manifold is introduced. This relation furthermore allows to extend the concept of symmetry operation to the tangent bundle; this extension furnishes, particularly in the euclidean case, a very practical way of representing symmetry operations of space groups completely devoid of any dependence on an origin, or, in other words, in which each and every point may be considered the origin. The investigation of the group of translations having being completed, the study of the linear parts of the isometries comes naturally. Based on the fact that the set of linear parts possesses the structure of a Lie group, several results are proven in a rigorous manner, such as the fact that a rotation angle of π/3 is incompatible with a three-dimensional cubic lattice. Procedures for determining different crystal systems in function of the type of rotation are laid out by way of the study of orthogonal matrices and their relation to the matrix associated with the type of system. Finally, the description of a crystal by its diffraction patterns is taken on. It is shown that the general aspect of such a pattern is directly linked to the action of that free and finite module of the Lie algebra of translations which generates a lattice on a manifold. In the case of modulated crystals, it is demonstrated that the appearance of supplementary spots is caused by the geometry, i.e. by the parametrisation of the manifold in which the crystal exists and not by the action of the module in the Lie algebra. Thus, there exists a neat separation: the geometrical aspect on the one hand, and the action of the group on the other. As the last topic, other ways of interpreting the diffraction pattern of a modulated structure are laid out in order to argue that mere experimental data do not warrant the uniqueness of a model. The goal of this study is by no means an attempt at overthrowing existing structural models such as the superspace-formalism or at revolutionising the methods for determining structures, but is rather aimed at sustaining that the definition of certain notions becomes thoroughly natural within the appropriate mathematical framework, and, that the term aperiodicity assigned to modulated structures no longer has a true meaning.

EPFL2009

David H. Templeton, 1920-2010

Gervais Chapuis

Acta Crystallographica Section A: Foundations of Crystallography covers theoretical and fundamental aspects of the structure of matter. The journal is the prime forum for research in diffraction physics and the theory of crystallographic structure determination by diffraction methods using X-rays, neutrons and electrons. The structures include periodic and aperiodic crystals, and non-periodic disordered materials, and the corresponding Bragg, satellite and diffuse scattering, thermal motion and symmetry aspects. Spatial resolutions range from the subatomic domain in charge-density studies to nanodimensional imperfections such as dislocations and twin walls. The chemistry encompasses metals, alloys, and inorganic, organic and biological materials. Structure prediction and properties such as the theory of phase transformations are also covered.