Читать книгу Solid State Chemistry and its Applications - Anthony R. West - Страница 17

Crystals are built up of regular arrangements of atoms in three dimensions; these arrangements can be represented by a repeat unit or motif called the unit cell. The unit cell is defined as the smallest repeating unit which shows the full symmetry of the crystal structure. Let us see exactly what this means, first in two dimensions. A section through the NaCl structure is shown in Fig. 1.1(a); possible repeat units are given in (b) to (e). In each, the repeat unit is a square and adjacent squares share edges and corners. Adjacent squares are identical, as they must be by definition; thus, all the squares in (b) have Cl⁻ ions at their corners and centres. The repeat units in (b), (c) and (d) are all of the same size and, in fact, differ only in their relative position. The choice of origin of the repeat unit is to some extent a matter of personal taste, even though its size, shape and orientation are fixed. The repeat unit of NaCl is usually chosen as (b) or (c) rather than (d) because it is easier to draw and visualise the structure as a whole if the repeat unit contains atoms or ions at special positions such as corners and edge centres. Another guideline is that usually the origin is chosen so that the symmetry of the structure is evident (Section 1.3).

In the hypothetical case that two‐dimensional (2D) crystals of NaCl could form, the repeat unit shown in (e), or its equivalent with Cl at the corners and Na in the middle, would be the correct unit. Comparing (e) and, for example, (c), both repeat units are square and show the 2D symmetry of the structure; as the units in (e) are half the size of those in (c), (e) would be preferred according to the above definition of the unit cell. In three dimensions, however, the unit cell of NaCl is based on (b) or (c), rather than (e) because only they show the cubic symmetry of the structure (see later).

Figure 1.1 (a) Section through the NaCl structure, showing (b–e) possible repeat units and (f) incorrect units.

In (f) are shown two examples of what is not a repeat unit. The top part of the diagram contains isolated squares whose area is one‐quarter of the squares in (c). It is true that each square in (f) is identical but it is not permissible to isolate unit cells or areas from each other, as happens here. The bottom part of the diagram contains units that are not identical; thus square 1 has Na in its top right corner whereas 2 has Cl in this position.

The unit cell of NaCl in three dimensions is shown in Fig. 1.2; it contains Na at the corner and face centre positions with Cl at the edge centres and body centre. Each face of the unit cell looks like the unit area shown in Fig. 1.1(c). As in the 2D case, the choice of origin is arbitrary; an equally valid unit cell could be chosen in which Na and Cl are interchanged. The unit cell of NaCl is cubic. The three edges: a, b and c are equal in length. The three angles: α (between b and c), β (between a and c) and γ (between a and b) are all 90°. A cubic unit cell also possesses certain symmetry elements and these, together with the shape define the cubic unit cell.

Figure 1.2 Cubic unit cell of NaCl, a = b = c.

Table 1.1 The seven crystal systems

Crystal system	Unit cell shapeb	Essential symmetry	Allowed lattices
Cubic	a = b = c, α = β = γ = 90°	Four threefold axes	P, F, I
Tetragonal	a = b ≠ c, α = β = γ = 90°	One fourfold axis	P, I
Orthorhombic	a ≠ b ≠ c, α = β = γ = 90°	Three twofold axes or mirror planes	P, F, I, A (B or C)
Hexagonal	a = b ≠ c, α = β = 90°, γ = 120°	One sixfold axis	P
Trigonal (a)	a = b ≠ c, α = β = 90°, γ = 120°	One threefold axis	P
Trigonal (b)	a = b = c, α = β = γ ≠ 90°	One threefold axis	R
Monoclinica	a ≠ b ≠ c, α = γ = 90°, β ≠ 90°	One twofold axis or mirror plane	P, C
Triclinic	a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90°	None	P

^a Two settings of the monoclinic cell are used in the literature, the most commonly used one given here, with b as the unique axis and the other with c defined as the unique axis: a ≠ b ≠ c, α = β = 90°, γ ≠ 90°.

^b The symbol ≠ means ‘not necessarily equal to’. Sometimes, crystals possess pseudo‐symmetry. For example, a unit cell may be geometrically cubic but not possess the essential symmetry elements for cubic symmetry; the true symmetry is then lower, perhaps tetragonal.

The seven crystal systems listed in Table 1.1 and shown in Fig. 1.3 are the seven independent unit cell shapes that are possible in three‐dimensional (3D) crystal structures. Six of these unit cell shapes are closely inter‐related and are either cubic or can be derived by distorting a cube in various ways, as shown in Fig. 1.3(b).

Thus, if one axis, c, is of different length to the others, the shape is tetragonal; if all three axes are different, the shape is orthorhombic. If, now, one of the angles, β, is not 90°, the shape is monoclinic, whereas if all three angles differ from 90°, the shape is triclinic. Finally, if the cube is stretched, or compressed, along a body diagonal so that all three angles remain equal, but different from 90°, the shape is trigonal.

The remaining unit cell shape is hexagonal. A hexagonal‐shaped box is shown in Fig. 1.3(a) and discussed later with reference to Fig. 1.21, but the true unit cell is only one‐third of this size, as shown.

Although it is common practice to describe unit cells by their shapes, it is more correct to describe them by the presence or absence of symmetry. Thus, for example, if a unit cell has four intersecting threefold axes, it must be cubic in shape; the reverse does not necessarily apply and the unit cell could be fortuitously cubic but not have the threefold symmetries in the atomic arrangements. The essential symmetry for each crystal system is given in the third column of Table 1.1. Let us deal next with symmetry.