Читать книгу Solid State Chemistry and its Applications - Anthony R. West - Страница 18

Symmetry is most easily defined using examples. Consider the silicate tetrahedron shown in Fig. 1.4(a). If it is rotated about an axis passing along the vertical Si–O bond, then every 120° the tetrahedron finds itself in an identical position. Effectively, the three basal oxygens change position with each other every 120°. During a complete 360° rotation, the tetrahedron passes through three identical positions. The fact that different (i.e. >1) identical orientations are possible means that the SiO₄ tetrahedron possesses symmetry. The axis about which the tetrahedron may be rotated is called a rotation axis; it is an example of a symmetry element. The process of rotation is an example of a symmetry operation.

Figure 1.3 (a) The seven crystal systems and their unit cell shapes; (b) five of the seven crystal systems can be derived from cubic by structural distortions.

Figure 1.4 (a) Threefold and (b) twofold rotation axes; (c) the impossibility of forming a complete layer of pentagons; (d) a complete layer of hexagons.

The symmetry elements that are important in crystallography are listed in Table 1.2. There are two nomenclatures for labelling them, the Hermann–Mauguin system used in crystallography and the Schönflies system used in spectroscopy. Ideally, there would be only one system which everybody uses, but this is unlikely to come about since (a) both systems are very well established, (b) crystallographers require elements of space symmetry that spectroscopists do not and vice versa, (c) spectroscopists use a more extensive range of point symmetry elements than crystallographers.

The symmetry element described above for the silicate tetrahedron is a rotation axis, with symbol n. Rotation about this axis by 360/n degrees gives an identical orientation and the operation is repeated n times before the original configuration is regained. In this case, n = 3 and the axis is a threefold rotation axis. The SiO₄ tetrahedron possesses four threefold rotation axes, one in the direction of each Si–O bond.

When viewed from another angle, SiO₄ tetrahedra possess twofold rotation axes [Fig. 1.4(b)] which pass through the central Si and bisect the O–Si–O bonds. Rotation by 180° leads to indistinguishable orientations of the tetrahedra. The SiO₄ tetrahedron possesses three of these twofold axes.

Crystals may display rotational symmetries 2, 3, 4 and 6. Others, such as n = 5, 7, are never observed in 3D crystal structures based on a regular periodic repetition of the unit cell and its contents. This is shown in Fig. 1.4(c), where a fruitless attempt has been made to pack pentagons to form a complete layer; thus, individual pentagons have fivefold symmetry but the array of pentagons does not. For hexagons with sixfold rotation axes (d), a complete layer is easily produced; both the individual hexagons and the overall array exhibit sixfold symmetry. This is not to say that molecules that have pentagonal symmetry, n = 5, cannot exist in the crystalline state. They can, of course, but their fivefold symmetry cannot be exhibited by the crystal as a whole.

Table 1.2 Symmetry elements

	Symmetry element	Hermann–Mauguin symbols (crystallography)	Schönflies symbols (spectroscopy)
Point symmetry	Mirror plane	m	σ v, σ h
	Rotation axis	n = 2, 3, 4, 6	C n (C 2, C 3, etc.)
	Inversion axis		–
	Alternating axisa	–	S n (S 1, S 2, etc.)
	Centre of symmetry		i
Space symmetry	Glide plane	a, b, c, d, n	–
	Screw axis	21, 31, etc.	–

^a The alternating axis is a combination of rotation (n‐fold) and reflection perpendicular to the rotation axis. It is little used in crystallography.