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Publication# The role of the tangent bundle for symmetry operations and modulated structures

Abstract

An equivalence relation on the tangent bundle of a manifold is defined in order to extend a structure (modulated or not) onto it. This extension affords a representation of a structure in any tangent space and that in another tangent space can easily be derived. Euclidean symmetry operations associated with the tangent bundle are generalized and their usefulness for the determination of the intrinsic translation part in helicoidal axes and glide planes is illustrated. Finally, a novel representation of space groups is shown to be independent of any origin point.

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Structure

A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buil

Tangent space

In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics

Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is

A new method for space-group determination is described. It is based on a symmetry analysis of the structure-factor phases resulting from a structure solution in space group P1. The output of the symmetry analysis is a list of all symmetry operations compatible with the lattice. Each symmetry operation is assigned a symmetry agreement factor that is used to select the symmetry operations that are the elements of the space group of the structure. On the basis of the list of the selected operations the complete space group of the structure is constructed. The method is independent of the number of dimensions, and can also be used in solution of aperiodic structures. A number of cases are described where this method is particularly advantageous compared with the traditional symmetry analysis.

2008The 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.

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.