the conditions for two planes (h1k1l1) and (h2k2l2) to lie in the same zone [uvw], that by multiplying Eq. (1.11) by a number m and Eq. (1.12) by a number n and adding them, we have:
(1.20)
Therefore the plane (mh1 + nh2, mk1 + nk2, ml1 + nl2) also lies in [uvw]. In other words, the indices formed by taking linear combinations of the indices of two planes in a given zone provide the indices of a further plane in that same zone. In general m and n can be positive or negative. If, however, m and n are both positive, then the normal to the plane under consideration must lie between the normals of (h1k1l1) and (h2k2l2): we will revisit this result in Section 2.2.
1.4 Symmetry Operators6
The symmetrical arrangement of atoms in crystals is described formally in terms of operations of symmetry. The symmetry arises because an atom or group of atoms is repeated in a regular way to form a pattern. Any operation of repetition can be described in terms of one of the following four different types of pure symmetry element or symmetry operator.
1.4.1 Translational Symmetry
This describes the fact that similar atoms in identical surroundings are repeated at different points within the crystal. Any one of these points can be brought into coincidence with another by an operation of translational symmetry. For instance, in Figure 1.1a the carbon atoms at O, Y, N and Q occupy completely similar positions. We use the idea of the lattice to describe this symmetry. The lattice is a set of points each with an identical environment which can be found by inspection of the crystal structure, as in the example in Figure 1.1b. We can define the arrangement of lattice points in a three‐dimensional crystal by observing that the vector r joining any two lattice points (or the operation of translational symmetry bringing one lattice point into coincidence with another) can always be written as:
(1.21)
where u, v, w are positive or negative integers, or equal to zero. Inspection of Figure 1.11 shows that to use this description we must be careful to choose a, b and c so as to include all lattice points. We do this by, say, making a the shortest vector between lattice points in the lattice, or one of several shortest ones. We then choose b as the shortest not parallel to a and c as the shortest not coplanar with a and b. Thus a, b and c define a primitive unit cell of the lattice in the same sense as in Section 1.1. Only one lattice point is included within the volume a · [b × c],7 which is the volume of the primitive unit cell; under these circumstances a, b and c are called the lattice translation vectors.
Figure 1.11 Translation symmetry in a crystal
1.4.2 Rotational Symmetry
If one stood at the point marked H in Figure 1.1a and regarded the surroundings in a particular direction, say that indicated by one of the arrows, then on turning through an angle of 60° = 360°/6 the outlook would be identical. We say an axis of sixfold rotational symmetry passes normal to the paper through the point H. Similarly, at O′ an axis of threefold symmetry passes normal to the paper, since an identical outlook is found after a rotation of 360°/3 = 120°. A crystal possesses an n‐fold axis of rotational symmetry if it coincides with itself upon rotation about the axis of 360°/n = 2π/n radians. In crystals, axes of rotational symmetry with values of n equal to one, two, three, four and six are the only ones found which are compatible with translational symmetry. These correspond to repetition every 360°, 180°, 120°, 90° and 60° and are called monad, diad, triad, tetrad and hexad axes, respectively. The reasons for these limitations on the value of n are explained in Section 1.5.
1.4.3 Centre of Symmetry
A centre of symmetry is a point where inversion through that point produces an identical arrangement. The operation of inversion moves the point with coordinates (x, y, z) to the position (−x, −y, −z). For instance, in Figure 1.3 (see also Figure 3.17), if we stood at the centre of the unit cell (coordinates
1.4.4 Reflection Symmetry
The fourth symmetry is reflection symmetry. The operation is that of reflection in a mirror. Mirror symmetry relates, for example, our left and right hands. The dotted plane running normal to the paper, marked m in Figure 1.12, reflects object A to A′, and vice versa. A′ cannot be moved about in the plane of the paper and made to superpose on A. The dotted plane is a mirror plane. It should be noted that the operation of inversion through a centre of symmetry also produces a right‐handed object from a left‐handed one.
Figure 1.12 Reflection symmetry
