Three-dimensional periodic symmetry space groups

Three-dimensional periodic symmetry space groups... 

Normally, solids are crystalline, i.e. they have a three-dimensional periodic order with three-dimensional translational symmetry. However, this is not always so. Aperiodic crystals do have a long-distance order, but no three-dimensional translational symmetry. In a formal (mathematical) way, they can be treated with lattices having translational symmetry in four- or five-dimensional space , the so-called superspace their symmetry corresponds to a four- or five-dimensional superspace group. The additional dimensions are not dimensions in real space, but have to be taken in a similar way to the fourth dimension in space-time. In space-time the position of an object is specified by its spatial coordinates x, y, z the coordinate of the fourth dimension is the time at which the object is located at the site x, y, z. 

In contrast to discrete molecules, crystals have a lattice structure exhibiting three-dimensional periodicity. As a result, we need to consider additional symmetry elements that apply to an infinitely extended object, namely the translations, screw axes, and glide planes. Chapters 9 and 10 introduce the concept and nomenclature of space groups and their application in describing the structures of crystals, as well as a survey of the basic inorganic crystalline materials. 

Objects or patterns which are periodic in one, two, and three directions will have one-, two-, and three-dimensional space groups, respectively. The dimensionality of the object/pattem is merely a necessary but not a satisfactory condition for the dimensionality of their space groups. We shall first describe a planar pattern after Budden [3] in order to get the flavor of space-group symmetry. Also, some new symmetry elements will be introduced. Later in this chapter, the simplest one-dimensional and two-dimensional space groups will be presented. The next Chapter will be devoted to the three-dimensional space groups which characterize crystal structures. 

In a further development of detail, one can take into account how the atoms of the solid are distributed spatially. The issue of symmetry in context with a fixed point in the crystal, and the symmetry of Bravais lattices, has been addressed, but in order to describe the entire crystal the effects of two new types of symmetry operation must be included. A space group determined in this way describes the spatial symmetry of the crystal. By definition, a crystallographic space group is the set of geometrical symmetry operations that take a three-dimensional periodic crystal into itself The set of operations that make up the space group must form a group in the mathematical sense and must include the primitive lattice translations as well as other symmetry operations. 

Among crystals with stacking faults the lack of a periodic order is restricted to one dimension this is called a one-dimensional disorder. If only a few layer positions occur and all of them are projected into one layer, we obtain an averaged structure. Its symmetry can be described with a space group, albeit with partially occupied atomic positions. The real symmetry is restricted to the symmetry of an individual layer. The layer is a three-dimensional object, but it only has translational symmetry in two dimensions. Its symmetry is that of a layer group there exist 80 layer-group types. 

Periodic repclitions of a space lattice cell in three dimensions from the original cell vvill completely partition space without overlapping or omissions. El is possible to develop a limited number of such three-dimensional patterns. Bravais. in 1848. demonsirated geometrically that there were but fourteen types of space lattice cells possible, and that these fourteen types could be subdivided into six groups called systems. Each system may be distinguished hy symmetry features, which can be related lo four symmetry elements ... 

Crystals of macromolecules, like those shown in Figure 3.1, are like crystals of all other kinds. They are precisely ordered three-dimensional arrays of molecules that may be characterized by a concise set of determinants that exactly define the disposition and periodicity of the fundamental units of which they are composed. The set of parameters is comprised of three elements. These define the symmetry properties, the repetitive and periodic features, and the distribution of atoms in the repeating unit. The properties may be separated and understood by considering how a crystal can be developed as a three-dimensional form from a basic building block (the asymmetric unit), by the application of symmetry (the space group), and translation (the unit cell, or lattice). As illustrated in Figure 3.2, this can be accomplished in four stages. 

In terms of diffraction properties and structural properties, a commensurate inclusion compound behaves as a conventional crystal. The periodicities of the host and guest molecules within the inclusion compound are described by a common three-dimensional (3D) periodic lattice, and the symmetry of the structure is described by a conventional 3D space group. Correspondingly, all diffractionmdLXima. in the diffraction pattern are described by a single 3D periodic reciprocal lattice. As discussed in more detail below, the inability to rationalize all diffraction maxima in the diffraction pattern of an inclusion compound by a single 3D periodic reciprocal lattice often provides the first indication that the structure may be incommensurate. 

A solid surface is intrinsically an imperfection of a crystalline solid by destroying the three-dimensional (3D) periodicity of the structure. That is, the unit cell of a crystal is usually chosen such that two vectors are parallel to the surface and the third vector is normal or oblique to the surface. Since there is no periodicity in the direction normal or oblique on the surface, a surface has a 2D periodicity that is parallel to the surface. By considering the symmetry properties of 2D lattices, one obtains the possible five 2D Bravais lattices shown in Figure 2. The combination of these five Bravais lattices with the 10 possible point groups leads to the possible 17 2D space groups. The symmetry of the surface is described by one of these 17 2D symmetry groups. 

In plane-wave calculations the periodic (repeated) slab model is used. In this model the slab is periodically repeated along the normal to the slab surface plane. This repeating of the slab restores the 3D periodicity of the surface model. The symmetry group of the periodic slab is a space group (Chap. 2) — three-dimensional group with... 

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