Fourteen Bravais lattices

Bravais showed in 1850 that all three-dimensional lattices can be classified into 14 distinct types, namely the fourteen Bravais lattices, the unit cells of which are displayed in Fig. 9.2.3. Primitive lattices are given the symbol P. The symbol C denotes a C face centered lattice which has additional lattice points at the centers of a pair of opposite faces defined by the a and b axes likewise the symbol A or B describes a lattice centered at the corresponding A or B face. When the lattice has all faces centered, the symbol F is used. The symbol I is applicable when an additional lattice point is located at the center of the unit cell. The symbol R is used for a rhombohedral lattice, which is based on a rhombohedral unit cell (with a = b = c and a = ft = y 90°) in the older literature. Nowadays the rhombohedral lattice is generally referred to as a hexagonal unit cell that has additional lattice points at (2/3,1 /3, /s) and (V3,2/3,2/3) in the conventional obverse setting, or ( /3,2/3, ) and (2/3, /3,2/3) in the alternative reverse setting. In Fig. 9.2.3 both the primitive rhombohedral (.R) and obverse triple hexagonal (HR) unit cells are shown for the rhombohedral lattice. 

---

A.2 The fourteen Bravais lattices and seven crystal systems Refer to Figs. A.2.1 and Table A.2.1. 

Figure 1.1 The fourteen Bravais lattices. The numbers 2, 3 etc. designate the symmetry axes and the letters P, F, I and C designate the lattice type.

Fig. 1.3 The fourteen Bravais lattice unit cells and associated lattice symbols.

Local coordination polyhedra the Jensen notation 7 Table 1.2 The fourteen Bravais lattices in three dimensions... 

Fig. 3.28 The fourteen Bravais lattices grouped according to the seven crystal systems.

The fourteen Bravais lattices are divided into seven crystal systems. The term system indicates reference to a suitable set of axes that bear specific relationships, as illustrated in Table 9.2.1. For example, if the axial lengths take arbitrary values and the interaxial angles are all right angles, the crystal system... 

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. 

Table IR-3.1 Pearson symbols used for the fourteen Bravais lattices...

The fourteen Bravais lattices are described in Table 2-1 and illustrated in Fig. 2-3, where the symbols P, F, /, etc., have the following meanings. We must first distinguish between simple, or primitive, cells (symbol P or R) and nonprimitive cells (any other symbol) primitive cells have only one lattice point per cell while nonprimitive have more than one. A lattice point in the interior of a cell belongs to that cell, while one in a cell face is shared by two cells and one at a corner is shared by eight. The number of lattice points per cell is therefore given by... 

Any of the fourteen Bravais lattices may be referred to a primitive unit cell. For example, the face-centered cubic lattice shown in Fig. 2-7 may be referred to the primitive cell indicated by dashed lines. The latter cell is rhombohedral, its axial angle a is 60°, and each of its axes is l/ /2 times the length of the axes of the cubic cell. Each cubic cell has four lattice points associated with it, each rhombohedral cell has one, and the former has, correspondingly, four times the volume of the latter. Nevertheless, it is usually more convenient to use the cubic cell rather than the rhombohedral one because the former immediately suggests the cubic symmetry which the lattice actually possesses. Similarly, the other centered nonprimitive cells listed in Table 2-1 are preferred to the primitive cells possible in their respective lattices. 

Why then do the centered lattices appear in the list of the fourteen Bravais lattices If the two cells in Fig. 2-7 describe the same set of lattice points, as they do, why not eliminate the cubic cell and let the rhombohedral cell serve instead The answer is that this cell is a particular rhombohedral cell with an axial angle a of 60°. In the general rhombohedral lattice no restriction is placed on the angle a the result is a lattice of points with a single 3-fold symmetry axis. When a becomes equal to 60°, the lattice has four 3-fold axes, and this symmetry places it in the cubic system. The general rhombohedral cell is still needed. 

The Fourteen Bravais Lattices Periodic Table of the Body Centered Cubic Elements Periodic Table of the Face Centered Cubic Elements Periodic Table of the Hexagonal Close Packed Elements Periodic Table of the Hexagonal Elements Structure of Ceramics... 

Fig. 2.4 Fourteen Bravais lattices (a) triclinic, (b) monoclinic, (c) orthorhombic, (d) tetragonal, (e) trigonal, (f) hexagonal, (g) cubic...

Fourteen Bravais lattices are divided into 7 crystallographic systems that are characterized by some special magnitude of lattice parameters. See Table 2.2 and Fig. 2.4. 

That of the crystallographic axes, which is related to the seven crystal systems. That of unit-cell topology, which is related to the fourteen Bravais lattices or... 

Figure 28.2 The Unit Cells of the Fourteen Bravais Lattices.

---