Читать книгу Introduction to Solid State Physics for Materials Engineers - Emil Zolotoyabko - Страница 13

Across this book, we will focus on physical properties of regular crystals, amorphs and quasicrystals being out of our scope here. Thinking on conventional crystals, we first keep in mind their translational symmetry. As we already mentioned, the long-range periodic order in crystals leads to translational symmetry, which is commonly described in terms of Bravais lattices (named after French crystallographer Auguste Bravais):

(1.1)

The nodes, r_s, of Bravais lattice are produced by linear combinations of three non-coplanar vectors, a₁, a₂, a₃, called translation vectors. The integer numbers in Eq. (1.1) can be positive, negative, or zero. Atomic arrangements within every crystal can be described by the set of analogous Bravais lattices.

Classification of Bravais lattices is based on the relationships between the lengths of translation vectors, |a₁| = a, |a₂| = b, |a₃| = c and the angles, α, β, γ, between them. In fact, all possible types of Bravais lattices can be attributed to seven symmetry systems:

Triclinic: a ≠ b ≠ c and α ≠ β ≠ γ;

Monoclinic: a ≠ b ≠ c and α = β = 90°, γ ≠ 90°; in this setting, angle γ is between translation vectors a₁ (|a₁| = a) and a₂ (|a₂| = b); whereas the angles α and β are, respectively, between translation vectors a₂^a₃ and a₁^a₃;

Orthorhombic : a ≠ b ≠ c and α = β = γ = 90°;

Tetragonal: a = b ≠ c and α = β = γ = 90°;

Cubic: a = b = c and α = β = γ = 90°;

Rhombohedral: a = b = c and α = β = γ ≠ 90°;

Hexagonal: a = b ≠ c and α = β = 90°, γ = 120°.

A parallelepiped built by the aid of vectors a₁, a₂, a₃ is called a unit cell and is the smallest block, which being duplicated by the translation vectors allows us to densely fill the 3D space without voids.

Translational symmetry, however, is only a part of the whole symmetry in crystals. Atomic networks, described by Bravais lattices, also possess the so-called local (point) symmetry, which includes lattice inversion with respect to certain lattice points, mirror reflections in some lattice planes, and lattice rotations about certain rotation axes (certain crystallographic directions). After application of all these symmetry elements, the lattice remains invariant. Furthermore, rotation axes are defined by their order, n. The latter, in turn, determines the minimum angle, , after rotation by which the lattice remains indistinguishable with respect to its initial setting (lattice invariance). In regular crystals, the permitted rotation axes, i.e. those matching translational symmetry (see Appendix 1.A), are twofold (180°-rotation, n = 2), threefold (120°-rotation, n = 3), fourfold (90°-rotation, n = 4), and sixfold (60°-rotation, n = 6). Of course, onefold, i.e. 360°-rotation (n = 1), is a trivial symmetry element existing in every Bravais lattice. The international notations for these symmetry elements are: – for inversion center, m – for mirror plane, and 1, 2, 3, 4, 6 – for respective rotation axes. We see that fivefold rotation axis and axes of the order, higher than n = 6, are incompatible with translational symmetry (see Appendix 1.A).

To deeper understand why some rotation axes are permitted, while others not, let us consider the covering of the 2D space by regular geometrical figures, having n equal edges and central angle, (Figure 1.4). Correspondingly, the angle, δ, between adjacent edges is:

(1.2)

To produce a pattern without voids by using these figures, we require that the full angle around each meeting point, M, defined by p adjacent figures, should be 360°, i.e. p · δ = 360°. Therefore, using Eq. (1.2) yields:

(1.3)

or

(1.4)

Figure 1.4 Dense filling of 2D space by regular geometrical figures.

Finally, we obtain:

(1.5)

It follows from Eq. (1.5) that there is a very limited set of regular figures (with 2 < n ≤ 6) useful for producing periodic patterns, which fill the 2D space with no voids (i.e. providing integer numbers, p). These are hexagons (n = 6, p = 3, ϕ = 60°), squares (n = 4, p = 4, ϕ = 90°), and triangles (n = 3, m = 6, ϕ = 120°). Based on the value of central angle, ϕ, these regular figures possess the sixfold, fourfold, and threefold rotation axes, respectively. Since they are related to regular geometrical figures, these rotation axes are called high-symmetry elements. Regarding the twofold axis, it fits the symmetry of the parallelogram, which also can be used for filling the 2D space without voids but does not represent a regular geometrical figure. For this reason, the twofold rotation axis is classified as a low symmetry element (together with inversion center, , and mirror plane, m). It also comes out from Eq. (1.5), that regular figures with fivefold rotation axis (n = 5), as well as with rotation axes higher than n > 6, are incompatible with translational symmetry, i.e. cannot be used for producing periodic patterns without voids since parameter, p, is not an integer number.

In the absence of the long-range translational symmetry, however, as in quasicrystals, one can find additional rotation axes, e.g. fivefold ( = 72°), as for 2D construction shown in Figure 1.3 or for icosahedral symmetry in 3D. The latter can be found in two Platonic bodies: regular icosahedrons and dodecahedrons. Regular dodecahedron has 12 pentagonal faces and 20 vertices, in each of them three faces meet (Figure 1.5). Therefore, the fivefold axes are normal to the pentagonal faces. In contrast, regular icosahedron has 20 triangular faces and 12 vertices, in each of them five faces meet (Figure 1.6). Therefore, the fivefold axes connect the body center and each vertex. Note that regular pentagon (plane figure) has central angle 72° and is characterized by the so-called golden ratio τ (the ratio between the pentagon diagonal, d_p, and pentagon edge, a_p, see Figure 1.7):

(1.6)

Figure 1.5 Dodecahedron sculpted by 12 pentagonal faces.

Figure 1.6 Icosahedron sculpted by 20 triangular faces.

Figure 1.7 Regular pentagon with edges equal a_p and diagonals equal d_p. The ratio, , is called the golden ratio (Eq. (1.6)).

which is of great importance to the quasicrystal diffraction conditions (described later in this chapter).

Permitted combinations of local symmetry elements (totally 32 in regular crystals) are called point groups. A set of different crystals, possessing the same point group symmetry, form certain crystal class. Point group symmetry is responsible for anisotropy of physical properties in crystals, as explained in more detail further in this chapter.

Figure 1.8 Unit cells of the following side-centered Bravais lattices: A-type (a), B-type (b), C-type (c). Translation vectors, a₁, a₂, a₃, are indicated by dashed arrows.

Figure 1.9 Unit cells of the following centered Bravais lattices: (a) face-centered (F-type) and (b) body-centered (I-type). Translation vectors, a₁, a₂, a₃, are indicated by dashed arrows.

Bravais lattices defined by Eq. (1.1) are primitive (P) since they effectively contain only one atom per unit cell. However, in some symmetry systems, 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 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, GaAs is non-centro-symmetric crystal belonging to class and revealing significant piezoelectric effect.

Figure 1.11 The presence of inversion center (C) in diamond structure (a) and its loss (X) in zinc-blende structure (b). Dissimilar atoms are indicated by different colors.

Combining local symmetry elements with translations creates novel elements of spatial symmetry – glide planes and screw axes. Therefore, spatial symmetry is a combination of local (point) symmetry and translational symmetry. As a result, 32 point groups + 14 Bravais lattices produce 230 space groups describing all possible variants of crystal symmetry, associated with charge distributions, i.e. related to geometrical points and polar vectors. Magnetic symmetry, linked to magnetic moments (axial vectors, see Section 1.2), will be discussed in Chapter 11.