Bravais points

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. 

The common periodic structures displayed by surfaces are described by a two-dimensional lattice. Any point in this lattice is reached by a suitable combination of two basis vectors. Two unit vectors describe the smallest cell in which an identical arrangement of the atoms is found. The lattice is then constructed by moving this unit cell over any linear combination of the unit vectors. These vectors form the Bravais lattices, which is the set of vectors by which all points in the lattice can be reached. 

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 ... 

The rotational operations generate a total of 32 Point Groups derived from these s)mimetry operations on the 14 Bravais lattices. 

The combination of the point lattices constructed on the basis of the crystallographic systems with the possible centring translation results in the 14 so-called Bravais lattice type, illustrated in Fig. 3.4. Substituting (decorating) each lattice... 

Crystal family Symbol Crystal system Crystallographic point groups (crystal classes) Number of space groups Conventional coordinate system Bravais lattices... 

Figure 3.4. The crystal systems and the Bravais lattices illustrated by a unit cell of each. All the points which, within a unit cell, are equivalent to each other and to the cell origin are shown. Notice that, in the primitive lattices the unit cell edges are coincident with the smallest equivalence distances. For the rhombohedral lattice, described in terms of hexagonal axis, the symbol hR is used instead of a symbol such as rP. In the construction of the so-called Pearson symbol ( 3.6.3), oS and mS will be used instead of oC and mC.

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. 

The Bravais lattice can be identified, on some specific Zone-Axis Patterns, from the observation of the shift between the reflection net located in the ZOLZ and the one located in the FOLZ. This shift is easily observed by considering the presence or the absence of reflections on the mirrors. Thus, in the example given on figure 1, some reflections from the ZOLZ are present on the four mi, m2, m3 and mirrors. This is not the case in the FOLZ where reflections are present on the m3 and m4 mirrors but not on the mi and m2 mirrors. Simulations given in reference [2] allow to infer the Bravais lattice from such a pattern. It is pointed out that Microdiffraction is very well adapted to this determination due to its good angular resolution (the disks look like spots). 

Figure 4.8 The 14 Bravais lattices. Black circles represent atoms or molecules. P cells contain only one lattice point, while C- and /-centred cells contain two and / -centred cells contain four.

Bravais, A. (1849) Etudes crystallographiques. Part I. Du cristal considere comme un simple assemblage de points. Paris, 101-194... 

The problem of the variation in the surface energies of various crystal facets can be attacked from several points of view. Bravais first noted that those planes of a crystal which were most densely packed and were also separated most distantly from the neighbouring parallel plane were those which appeared most frequently in crystals he noted also that a closely packed surface was usually associated with a wide interplanar distance and vice versa. Later Willard Gibbs indicated that the most stable planes on a growing crystal were those possessing the least interfacial surface energy. 

If we combine the 32 crystal point groups with the 14 Bravais lattices we find 230 three-dimensional space groups that crystal structures can adopt (i.e., 230... 

A. Bravais, Les systemes formes par des pointes distribues regulierement sur un plan on dans I espace, J. Be. Polytech., XIX, 1850,1-128... 

Again, our first concern must be to see how many ways there are in which the translation vectors can be related to one another (relative lengths, angles between them) to give distinct, space-filling patterns of equivalent points. We have seen (Section 11.2) that in 2D there were only 5 distinct lattices. We shall now see that in 3D there are 14. These are often designated eponymously as the Bravais lattices and are shown in Figure 11.11, in the form of one unit cell of each. 

Now that we have enumerated all of the 3D lattices, the 14 Bravais lattices, we can look in more detail at their symmetries. First of all, it must be recognized that every lattice point is a center of symmetry. The translation vectors tx, t2, and t3 are entirely equivalent to tj, -t2, and -t3, respectively. Therefore, in determining the point symmetry at each lattice point (which is what symmetry of the lattice means) we must include the inversion operation and all its products with the other operations. 

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. 

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