Solid State Chemistry and its Applications. Anthony R. West

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       Figure 1.61 Two tetrahedra in a centrosymmetric arrangement.

In summary, therefore, the unit cell in space group P has two equivalent positions that lie inside the unit cell: x, y, z (position 1) and , , (the position at height с above 2″ in Fig. 1.60).

      Although only one centre of symmetry is necessary to generate the equivalent positions in P , many other centres of symmetry are created automatically. For example, the centre of symmetry at и arises because pairs of positions such as 1 and 2‴, 2 and 1‴, etc., are centrosymmetrically related through u. This may be seen from the diagram or may be proven by comparing coordinates of the three positions: positions 2‴ and 1 are equidistant from и and lie on a straight line that passes through u.

      The positions x, y, z and , _, are general positions and apply to any value of x, у, z between 0 and 1. In certain circumstances, x, y, z and , ½, coincide, for example, if x = y = z = ½. In this case, there is only one position, ½,½,½ which is a special position. Special positions arise when the general position lies on a symmetry element, in this case a centre of symmetry, as discussed for point groups in Section 1.18.3. The coordinates of the one‐fold special positions in P are, therefore, (0, 0, 0), (½, 0, 0), (0, ½, 0), (0, 0, ½), (½, ½, 0), (½, 0, ½), (0, ½, ½) and (½, ½, ½), and correspond to the corner, edge, face and body centres of the unit cell.

      The coordinates of the general and special positions for each space group are listed in International Tables for X‐ray Crystallography together with additional information, as shown in Table 1.30 for P . The general positions are listed first followed by the various special positions; for each, the first column gives the multiplicity, i.e. the number of equivalent positions. The second column is the so‐called Wyckoff notation in inverse alphabetical sequence; the logic behind this labelling scheme is that those positions at the bottom of the list have highest point symmetry and the symmetry may decrease on moving upwards through the list. The third column specifies the point symmetry of the positions. In this case, all the special positions coincide with a centre of symmetry whereas the general position, labelled as 2(i), has no point symmetry.

       1.18.5.2 Monoclinic C2

By common convention, the unique 2‐fold axis in a monoclinic unit cell is labelled b. Unfortunately, this is different to the use of с as the unique axis in tetragonal, trigonal and hexagonal cells but this usage for monoclinic cells is now so well established that it is unlikely to be altered. With b as the unique axis, the unit cell projects onto the xy plane as a rectangle (because γ = 90°), as shown in Fig. 1.62. Since β ≠ 90°, the z axis is not perpendicular to the plane of the paper but is inclined to the vertical.

      Table 1.30 Coordinates and labelling of equivalent positions in space group P

Number of positions	Wyckoff notation	Point symmetry	Coordinates of equivalent positions
2	I	1	xyz,
1	h		½, ½, ½
1	g		0, ½, ½
1	f		½, 0, ½
1	e		½, ½, 0
1	d		½, 0, 0
1	c		0, ½, 0
1	b		0, 0, ½
1	a		0, 0, 0

      The C‐centring in space group C2 means that if the Bravais lattice has a lattice point at the origin (with coordinates 0, 0, 0), it also has an equivalent lattice point in the middle of the side bounded by a and b, at ½, ½, 0. For any position x, y, z in this space group, there will, therefore, be an equivalent position at x + ½, y + ½, z. This C‐centring has no representation in the right‐hand diagram of Fig. 1.62 but can be seen in the left‐hand diagram; for example, positions 1 and 2 are related by the C‐centring.