Bravais elements

If atoms, molecules, or ions of a unit cell are treated as points, the lattice stmcture of the entire crystal can be shown to be a multiplication ia three dimensions of the unit cell. Only 14 possible lattices (called Bravais lattices) can be drawn in three dimensions. These can be classified into seven groups based on their elements of symmetry. Moreover, examination of the elements of symmetry (about a point, a line, or a plane) for a crystal shows that there are 32 different combinations (classes) that can be grouped into seven systems. The correspondence of these seven systems to the seven lattice groups is shown in Table 1. 

If we now apply rotadonal nnmetxy (Factor II given in 2.2.1) to the 14 Bravais lattices, we obtain the 32 Point-Groups which have the factor of symmetry imposed upon the 14 Bravais lattices. The symmetry elements that have been used are ... 

Electron Diffraction (CBED) and Large-Angle Convergent-Beam Electron Diffraction (LACBED) allow the identification of the crystal system, the Bravais lattice and the point and space groups. These crystallographic features are obtained at microscopic and nanoscopic scales from the observation of symmetry elements present on electron diffraction patterns. 

The complete charge array is built by the juxtaposition of this cell in three dimensions so that to obtain a block of 3 x 3 x 3 cells, the cluster being located in the central cell. In that case the cluster is well centered in an array of475 ions. Practically and for computational purposes, the basic symmetry elements of the space group Pmmm (3 mirror planes perpendicular to 3 rotation axes of order 2 as well as the translations of the primitive orthorhombic Bravais lattice) are applied to a group of ions which corresponds to 1/8 of the unit cell. The procedure ensures that the crystalline symmetry is preserved. 

Lattice type is the basis of calculation of reticular density in the Bravais empirical law. In lattice types, only the symmetry elements with no translation, i.e. the... 

Table 1.1 gives the structures of the elements at zero temperature and pressure. Each structure type is characterized by its common name (when assigned), its Pearson symbol (relating to the Bravais lattice and number of atoms in the cell), and its Jensen symbol (specifying the local coordination polyhedron about each non-equiyalent site). We will discuss the Pearson and Jensen symbols later in the following two sections. We should note,... 

We can now complete our answer to the question, What information is conveyed when we read that the crystal structure of a substance is monodime P2JC7" The structure belongs to the monoclinic crystal system and has a primitive Bravais lattice. It also possesses a two-fold screw axis and a glide plane perpendicular to it. The existence of these two elements of symmetry requires that there also be a center of inversion. The latter is not specifically included in the space group notation as it would be redundant. 

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 ... 

Symmetry is the fundamental basis for descriptions and classification of crystal structures. The use of symmetry made it possible for early investigators to derive the classification of crystals in the seven systems, 14 Bravais lattices, 32 crystal classes, and the 230 space groups before the discovery of X-ray crystallography. Here we examine symmetry elements needed for the point groups used for discrete molecules or objects. Then we examine additional operations needed for space groups used for crystal structures. 

Figure 7.10 shows the 14 three-dimensional Bravais lattices available for monoatomic solids. This means that, for a crystal consisting of only one atom, there are only 14 ways in which this crystal can fill space. In practice, the stable crystal structures of the chemical elements chose only a few of these lattices. 

The problem of combining the point groups with Bravais lattices to provide a finite number of three-dimensional space groups was worked out independently by Federov and by Schoenflies in 1890. Since the centred cells contain elements of translational symmetry new symmetry elements, not of the point-group type are generated in the process. 

The class symbols can be derived from the space group symbols by deleting the Bravais symbols (P, C, etc.), dropping all subscripts from screw axes (2i, 3i, 4i, etc. -> 2,3,4, etc.) and replacing all glide plane symbols by the mirror plane symbol, m. Thus I4i/acd becomes 4/mmm. A slash means perpendicularity of a rotational element and a reflection element. 

For example, think about the monoclinic point group m in the standard setting, where m is perpendicular to b (Table 1.8). According to Table 1.14, the following Bravais lattices are allowed in the monoclinic crystal system P and C. There is only one finite symmetry element (mirror plane m) to be considered for replacement with glide planes (a, b, c, n and d) ... 

The second position in the symbol is occupied by a standard notation of Bravais lattice. Thus, the first two elements in the Pearson s symbol are letters and they classify all available alloy structures according to 14 Bravais lattices, as shown in Table 6.1. 

Both Bravais lattices and the real crystals which are built up on them exhibit various kinds of symmetry. A body or structure is said to be symmetrical when its component parts are arranged in such balance, so to speak, that certain operations can be performed on the body which will bring it into coincidence with itself. These are termed symmetry operations. For example, if a body is symmetrical with respect to a plane passing through it, then reflection of either half of the body in the plane as in a mirror will produce a body coinciding with the other half. Thus a cube has several planes of symmetry, one of which is shown in Fig. 2-6(a). There are in all four macroscopic symmetry operations or elements reflection. 

It is now time to describe the structure of some actual crystals and to relate this structure to the point lattices, crystal systems, and symmetry elements discussed above. The cardinal principle of crystal structure is that the atoms of a crystal are set in space either on the points of a Bravais lattice or in some fixed relation to those points. It follows from this that the atoms of a crystal will be arranged periodically in three dimensions and that this arrangement of atoms will exhibit many of the properties of a Bravais lattice, in particular many of its symmetry elements. 

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