axis is the same as a mirror plane perpendicular to that axis.
Figure 1.7 Symmetry elements: (a) mirror plane; (b) centre of symmetry; (c) fourfold inversion axis. (d) The British comedian Harry Worth creating a mirror image of half of his body by reflection in a shop window.
1.2.5 Point symmetry and space symmetry
The symmetry elements discussed so far are elements of point symmetry. For each, at least one point stays unchanged during the symmetry operation, i.e. an atom lying on a centre of symmetry, rotation axis or mirror plane does not move during the respective symmetry operations. Finite‐sized molecules can only possess point symmetry elements, whereas crystals may have extra symmetries that include translation steps as part of the symmetry operation. These are elements of space symmetry, of which there are two types.
The screw axis combines translation and rotation; the atoms or ions in a crystal which possesses screw axes appear to lie on spirals about these axes. A schematic example is shown in Fig. 1.8(a). All you need to demonstrate a screw axis is a handful of coins which can be arranged on a flat surface with either their heads (H) or tails (T) facing upwards. The symbol for a screw axis, XY, indicates translation by the fraction Y/X of the unit cell edge in the direction of the screw axis together with simultaneous rotation by 360/X° about the screw axis. Thus, a 42 axis parallel to a involves translation by a/2 and rotation by 90°; this process takes place twice along a for every unit cell.
Figure 1.8 Arrangement of coins with heads (H) and tails (T) illustrating (a) a 21 screw axis parallel to a and (b) an a glide plane perpendicular to b; in each case, translation between symmetry‐related objects is by a/2.
The glide plane combines translation and reflection, as shown schematically in Fig. 1.8(b). Translation may be parallel to any of the unit cell axes (a, b, c), to a face diagonal (n) or to a body diagonal (d). The a, b, c and n glide planes all have a translation step of half the unit cell in that direction; by definition, the d glide has a translation step which is ¼ of the body diagonal. For the axial glide planes a, b and c, it is important to know both the direction of the translation and the reflection plane, e.g. an a glide may be perpendicular to b (i.e. reflection across the ac plane), Fig 1.8(b), or perpendicular to c.
Most crystal structures contain examples of space symmetry elements which are often difficult to visualise unless one has 3D models available. An example of both a screw axis and a glide plane in the same structure is shown in Fig 1.21 and discussed in Section 1.12. It is always amusing to look for symmetries in everyday objects. Next time you are in a car park, look at the arrangement of parking lots, especially if they are arranged diagonally, and see if they are displaced relative to each other in the form of a glide plane!
One difference between crystals and quasicrystals concerns space symmetry. Crystals, as a bare minimum, exhibit a periodic translation from one unit cell to the next. In many cases, translational symmetry elements, i.e. screw axes and glide planes, are also present. By contrast, quasicrystals do not exhibit space symmetry: they do not have a regular repeat unit, nor can translational symmetry elements be identified.
1.3 Symmetry and Choice of Unit Cell
The geometric shapes of the various crystal systems (unit cells) are listed in Table 1.1 and are shown in Fig. 1.3. These shapes do not define the unit cell; they are merely a consequence of the presence of certain symmetry elements.
A cubic unit cell is defined as one having four threefold symmetry axes. These run parallel to the cube body diagonals, one of which is shown in Fig. 1.9(a); an automatic consequence of this condition is that a = b = c and α = β = γ = 90°. The essential symmetry elements by which each crystal system is defined are listed in Table 1.1. In most crystal systems, additional symmetry elements are also present. For instance, cubic crystals may have many others, including three fourfold axes passing through the centres of each pair of opposite cube faces (a) and mirror planes in two sets of orientations (b, c).
The tetragonal unit cell, Table 1.1, has a single fourfold axis and may be regarded as a cube that is either squashed or elongated along one axis. Consequently, all threefold rotation axes and two of the fourfold axes are lost. An example is given by the structure of CaC2. This is related to the NaCl structure but, because the carbide ion is cigar‐shaped rather than spherical, one of the cell axes becomes longer than the other two, Fig. 1.10(a). A similar tetragonal cell may be drawn for NaCl; it occupies half the volume of the true, cubic unit cell, Fig. 1.10(b). The choice of a tetragonal unit cell for NaCl is rejected, however, because it does not show the full cubic symmetry, i.e. it does not show the threefold axes. A tetragonal unit cell has one axis, the unique axis, which is of a different length to the other two. By convention, this is chosen as the c axis, Table 1.1.
Figure 1.9 (a) Two‐, three‐ and fourfold axes and (b, c) mirror planes of a cube.
Figure 1.10 (a) Tetragonal unit cell of CaC2: note the cigar‐shaped carbide ions are aligned parallel to c; (b) relation between ‘tetragonal’ and cubic cells for NaCl; (c) derivation of a primitive trigonal unit cell for NaCl from the cubic cell.
The trigonal system is characterised by a single threefold axis. Its shape is derived from a cube by stretching or compressing the cube along one of its body diagonals, Fig. 1.10(c). The threefold axis parallel to this direction is retained but those along the other body diagonals are destroyed. All three cell edges remain the same length; all three angles stay the same but are not equal to 90°. It is possible to describe such a trigonal cell for NaCl with α = β = γ = 60° with Na at the corners and Cl in the body centre (or vice versa), but this is again unacceptable because NaCl has symmetry higher than trigonal.
The structure of NaNO3 is a trigonal distortion of the NaCl structure: instead of spherical Cl– ions, it has triangular nitrate groups. These lie on planes perpendicular to one of the unit cell body diagonals. This causes a compression along one body diagonal (or rather an expansion in the plane perpendicular to the diagonal). All fourfold symmetry axes and all but one of the threefold axes are destroyed.
The hexagonal crystal system is discussed later (Fig. 1.21).
The orthorhombic unit cell may be regarded as a shoebox in which
