Crystal symmetries Bravais lattices

Crystal system Bravais lattices Unit cell constraints Laue symmetry... 

Crystal system (Bravais lattices) defining symmetry elements Crystallographic point groups (molecular point groups )... 

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. 

These 14 Bravais Lattices are unique in themselves. If we arrange the crystal systems in terms of symmetry, the cube has the highest symmetry and the triclinic lattice, the lowest symmetry, as we showed above. The same hierarchy is maintained in 2.2.4. as in Table 2-1. The symbols used by convention in 2.2.4. to denote the type of lattice present are... 

Microdiffraction is the pertinent method to identify the crystal system, the Bravais lattices and the presence of glide planes [4] (see the chapter on symmetry determination). For the point and space group identifications, CBED and LACBED are the best methods [5]. 

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. 

In Section 11.4 the fourteen 3D lattices (Bravais lattices) were derived and it was shown that they could be grouped into the six crystal systems. For each crystal system the point symmetry of the lattice was determined (there being one point symmetry for each, except the hexagonal system that can have either one of two). These seven point symmetries are the highest possible symmetries for crystals of each lattice type they are not the only ones. 

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. 

It is not always possible to choose a unit cell which makes every pattern point translationally equivalent, that is, accessible from O by a translation a . The maximum set of translationally equivalent points constitutes the Bravais lattice of the crystal. For example, the cubic unit cells shown in Figure 16.2 are the repeat units of Bravais lattices. Because nt, n2, and w3 are integers, the inversion operator simply exchanges lattice points, and the Bravais lattice appears the same after inversion as it did before. Hence every Bravais lattice has inversion symmetry. The metric M = [a, a ] is invariant under the congruent transformation... 

When we consider crystal structures we usually think of the pattern and symmetry of the packing of the atoms, ions, or molecules in building the lattice based on X-ray crystallography. However, detailed descriptions of crystals and their classification are much older. The seven systems of crystals and the 32 classes of crystal symmetry were recognized by 1830. The 14 Bravais Lattices were presented by A. Bravais in 1848. 

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. 

The Seven Systems of Crystals are shown in Figure 2.2. The relationship between the trigonal and rhombohedral systems is shown in Figure B.la. The possibilities of body-centered and base-centered cells give the 14 Bravais Lattices, also shown in Figure 2.2. A face-centered cubic (fee) cell can be represented as a 60° rhombohedron, as shown in Figure B.lb. The fee cell is used because it shows the high symmetry of the cube. 

The 32 crystallographic point groups, first mentioned in Table 7.1, are now described in Table 7.8 (ordered by principal symmetry axes and also by the crystal system to which they belong). The 230 space groups of Schonflies and Fedorov were generated systematically by combining the 14 Bravais lattices with the intra-unit cell symmetry operations for the 32 crystallographic point... 

The only possible cells in two dimensions are oblique (p only), rectangular (p and c) and hexagonal (p). For each of the seven three-dimensional crystal systems primitive and centred cells can be chosen, but centring is not advantageous in all cases. In the case of triclinic cells no centred cell can have higher symmetry than the primitive and is therefore avoided. In all there are 14 different lattice types, known as the Bravais lattices Triclinic (P), Monoclinic (P,C), Orthorhombic (P,C,I,F), Trigonal (R), Tetragonal (P,I), and Cubic (P,I,F). 

FIGURE 7-1 The Seven Crystal Classes and Fourteen Bravais Lattices. The points shown are not necessarily individual atoms, but are included to show the necessary symmetry. 

Crystal system The seven crystal systems, classified in terms of their symmetry and corresponding to the seven fundamental shapes for unit cells consistent with the 14 Bravais lattices. 

A more exact procedure is to solve the Bom-von Karman equations of motions 38) to obtain frequencies as a function of the wave vector, q, for each branch or polarization. These will depend upon unit-cell symmetry and periodicity, force constants, and masses. Thus, for a simple Bravais lattice with identical atoms per unit cell, one obtains three phase-frequency relations for the three polarizations. For crystals having two atoms per unit cell, six frequencies are obtained for each value of the phase or wave vector. When these equations have been solved for a sufficient number of wave-vectors, g hco) can, in principle, be obtained by direct count . Thus, a recent calculation (13) of g to) based upon a normal-mode calculation that included intermolecular forces gave an improved fit to the specific heat data of Wunderlich, and showed additional peaks of 140, 90 and 60 cm in the frequency distribution. Even with this procedure, care must be exercised, since it has been shown that significant features of g k(o) may be rormded out. Topological considerations have shown that significant structure in g hco) vs. ho may arise from extreme or saddle points in the phase-frequency curves (38). 

The introduction of lattice centering makes the treatment of crystallographic symmetry much more elegant when compared to that where only primitive lattices are allowed. Considering six crystal families Table 1.12) and five types of lattices Table 1.13), where three base-centered lattices, which are different only by the orientation of the centered faces with respect to a fixed set of basis vectors are taken as one, it is possible to show that only 14 different types of unit cells are required to describe all lattices using conventional crystallographic symmetry. These are listed in Table 1.14, and they are known as Bravais lattices. ... 

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 ensemble of all equivalent positions for a space group is unique and may be considered the mathematical definition of the space group. It provides the basis for manipulating objects and points related by symmetry in a digital computer. Equivalent positions are another way of stating both the space group and the Bravais lattice of a crystal. 

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