Introduction to Solid State Physics for Materials Engineers. Emil Zolotoyabko

the same local symmetry will be held for centered Bravais lattices, in which the symmetry-related equivalent points are not only the corners (vertices) of the unit cell (as for primitive lattice), but also the centers of the unit cell faces or the geometrical center of the unit cell itself (Figures 1.8 and 1.9). Such lattices are conventionally called side-centered (A, B, or C), face-centered (F), and body-centered (I). In side-centered modifications of the type A, B, or C, additional equivalent points are in the centers of two opposite faces, being perpendicular, respectively, to the a₁-, a₂-, or a₃- translation vectors (Figure 1.8). In the face-centered modification, F, all faces of the Bravais parallelepiped (unit cell) are centered (Figure 1.9). For the cubic symmetry system, the F-centered Bravais lattice is called face-centered cubic (fcc). In the body-centered modification, I, the center of the unit cell is symmetry-equivalent to the unit cell vertices (Figure 1.9). For the cubic symmetry system, the I-modification of the Bravais lattice is called body-centered cubic (bcc). Accounting of centered Bravais lattices increases their total amount up to 14.

In some cases, the choice of Bravais lattice is not unique. For example, fcc lattice can be represented as rhombohedral one with a_R = a/ and α = 60° (Figure 1.10a). Rhombohedral lattice is a primitive one and comprises one atom per unit cell instead four atoms in the fcc unit cell. Similarly, bcc lattice can be represented in the rhombohedral setting with a_R = a/2 and α = 109.47° (Figure 1.10b). In this case, the rhombohedral lattice comprises one atom per unit cell instead two atoms in the bcc unit cell. We will widely use these settings in Chapter 2 considering the shapes of Brillouin zones. Minimizing number of atoms in the unit cell substantially reduces the calculation complexity of different physical properties in crystals.

Figure 1.10 Lattice translations (red arrows) in the rhombohedral setting of the fcc (a) and bcc (b) lattices.

Table 1.1 Summary of possible symmetries in regular crystals.

Crystal symmetry	Bravais lattice type	Crystal classes (point groups)
Triclinic	P	1,
Monoclinic	P; B, or C	m, 2, 2/m
Orthorhombic	P; A, B, or C; I; F	mm2, 222, mmm
Tetragonal	P; I	4, 422, , , 4/m, 4mm, 4/mmm
Cubic	P; I (bcc); F (fcc)	23, , 432, ,
Rhombohedral (trigonal)	P ( R )	3, 32, 3m, ,
Hexagonal	P	6, 622, , , 6/m, 6mm, 6/mmm

Symmetry systems, types of Bravais lattices, and distribution of crystal classes (point groups) among them are summarized in Table 1.1.

      The number of high-order symmetry elements, i.e. the threefold, fourfold, and sixfold rotation axes, which can simultaneously appear in a crystal, is also symmetry limited. For threefold rotation axis, this number may be one, in trigonal classes, or four, in cubic classes; for fourfold rotation axes – one in tetragonal classes or three in some cubic classes, while for sixfold rotation axis – only one in all hexagonal classes (see Appendix 1.A).

The presence or absence of an inversion center in a crystal is of upmost importance to many physical properties. For example, ferroelectricity and piezoelectricity (see Chapter 12) do not exist in centro-symmetric crystals, i.e. in those having inversion center. In this context, it is worth to note that any Bravais lattice is centro-symmetric. For primitive lattices, this conclusion follows straightforwardly from Eq. (1.1). Centered (non-primitive) Bravais lattices certainly do not refute this statement (Figures 1.8 and 1.9). However, only 11 crystal classes of total 32, in fact, are centro-symmetric. Even for high cubic symmetry, only two classes are centro-symmetric, i.e. and (Table 1.1). Evidently, the loss of an inversion center can happen in crystals, which are built of several Bravais lattices, their origins being shifted relative to each other. We stress that it is necessary, but not sufficient condition for the loss of inversion center. For illustration, let us consider Si (diamond structure) and GaAs (zinc blende or sphalerite structure) crystals. Both comprise two fcc lattices shifted relative to each other by one quarter of a space cube diagonal. The difference is that in silicon these sub-lattices are occupied by identical atoms (Si), whereas in GaAs – separately