The 14 Bravais Space Lattices

Auguste Bravais (1811-1863) first proposed the Miller-Bravais system for indices. Also, as a result of his analyses of the external forms of crystals, he proposed the 14 possible space lattices in 1848. His Etudes Cristallographiques, published in 1866, after his death, treated the geometry of molecular polyhedra. 

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... 

By such methods, the structures of crystals have been determined and all of them can be shown to possess space structures corresponding to one or another of the 14 Bravais lattices. See Figs. 3 and 4. 

Chapter 2 Crystals, Point Groups, and Space Groups Table 2.1. Crystal systems and the 14 Bravais Lattices. 

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... 

Figure 16 Representation of the 14 Bravais lattices of three-dimensional space...

Bravais in 1849 showed that there are only 14 ways that identical points can be arranged in space subject to the condition that each point has the same number of neighbors at the same distances and in the same directions.Moritz Ludwig Frankenheim, in an extension of this study, showed that this number, 14, could also be used to describe the total number of distinct three-dimensional crystal lattices.These are referred to as the 14 Bravais lattices (Figure 4.9), and they represent combinations of the seven crystal systems and the four lattice centering types (P, C, F, I). Rhombohedral and hexagonal lattices are primitive, but the letter R is used for the former. 

TABLE 4.3. The 14 Bravais Lattices, 32 Crystallographic Point Groups (Crystal Classes) and Some Space Groups. 

As stated above, the 32 point groups describe the symmetry of the unit cells, and the 14 Bravais lattices describe all possible arrangements of crystal lattices. To describe space symmetry, however, symmetry operations mentioned earlier are not enough. We must add three new symmetry operations ... 

Factor I gives rise to the 14 Bravais lattices Factor II generates the 32 point- groups Factor III creates the 232 space- groups... 

Bravais proved that if the contents of a unit cell have symmetry, the number of distinct types of space lattices becomes fourteen. These are the only lattices that can fill all space and are commonly termed the 14 Bravais lattices. Since there are seven crystal systems, it might be thought that by combining the seven crystal systems with the idea of a primitive lattice a total of seven distinct Bravais lattices would be obtained. However, it turns out that the trigonal and hexagonal lattices so constructed are equivalent, and therefore only six lattices can be formed in this way. These lattices, which are given the label P, define the primitive unit cells (or P-cells) in each case. 

A central tenet of materials science is that the behavior of materials (represented by their properties) is determined by their structure on the atomic and microscopic scales (Shackelford, 1996). Perhaps the most fundamental aspect of the structure-property relationship is to appreciate the basic skeletal arrangement of atoms in crystalline solids. Table 2.21 illustrates the fundamental possibilities, known as the 14 Bravais lattices. All crystalline structures of real materials can be produced by decorating the unit cell patterns of Table 2.21 with one or more atoms and repetitively stacking the unit cell structure through three-dimensional space. 

There are precisely 14 different topological ways of arranging equivalent points in an atomic array and this gives rise to the 14 Bravais lattices or space lattices, as it was Auguste Bravais in 1848 who first rigorously established that other suggested lattices were in fact identical to one of his own 14. These lattices are named by their crystal system followed by a symbol P, /, F, C or / always italicized) as indicated in Fig. 7 [1]. 

Figure 7 The 14 Bravais lattices or space lattices (see text for the definitions). (After Ref. 1.)...

Conclusion Therefore, all crystals have a space lattice that must be one of the 14 Bravais lattices and not more, as for these are the only ways in which indistinguishable points can occur uniquely in three dimensions of space. 

All crystal systems can be classified into one of the 14 Bravais lattices which can be subdivided into 32 crystal classes or point groups. If certain other translation operations that do not have point symmetry are considered, such as a translation combined with a mirror reflection (glide plane operation) or a translation combined with an n-fold rotation (screw axis), the 32 point groups can be subdivided into 230 possible space groups that completely describe the symmetry of all possible crystal systems. These are enumerated in the International Tables for Crystallography, vol A (Ed. Th. Hahn, 2006). 

There are seven different polyhedra with different symmetries that can fill all 3-D space. These polyhedra form the primitive cells of seven basic crystal systems. It is also possible to form an additional seven nonprimitive systems with higher symmetry by adding lattice points in the center or on the faces of the basic systems, thus forming the 14 Bravais lattices that describe all crystals. 

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