Introduction to Solid State Physics for Materials Engineers. Emil Zolotoyabko

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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 by Ga and As. In a result, Si is centro-symmetric (class ) that can be easily proved by setting inversion center at point (⅛,⅛,⅛), i.e. in the middle between the origins of two centro-symmetric fcc Si sub-lattices (Figure 1.11a). This recipe can hardly be used in case of GaAs since there is no symmetry operation that converts Ga to As (Figure 1.11b). Therefore, СКАЧАТЬ