Solid State Chemistry and its Applications. Anthony R. West

= ½ − z. This is shortened to ½– in Fig. 1.64.

      Consider now the 2‐fold axis parallel to x and at b = c = 0 (i.e. passing through the origin). This axis generates positions 3″ from 1 and 4‴ (its equivalent in the cell below) from 2′. With these two axes we have generated all four equivalent positions in this space group. The third axes, such as the 2₁ axis parallel to z, are automatically generated by the combined action of the other two axes and are not independent of them. This 2₁ axis relates, for example, positions 1 and 4‴, (i.e. translation of position 1 by c/2 followed by 180° rotation about с gives 4‴). Positions 2′ and 3″ are similarly related.

       1.18.5.5 Orthorhombic F222

The new feature of this space group is that it has a face centred lattice which, as can be seen from Fig. 1.65, leads to a considerable increase in the number of symmetry elements and equivalent positions. The basic symmetry elements are three intersecting 2‐fold axes, parallel to x, у and z and passing through the origin. Many other 2‐fold axes occur automatically, e.g. one intersecting the cell at a = , с = , parallel to b and another at a = , b = , parallel to c. Many 2₁ axes are also created, e.g. at a = 0, b = , parallel to с and at b = , c = 0, parallel to a.

      Figure 1.64 Orthorhombic space group P2221 (No 17); Coordinates of equivalent positions 4(e): x, y, z; , y, ½ − z; x, ; . Special positions with point symmetry 2, 2(a): x, 0, 0; , 0, ½; 2(b): x, ½, 0; _, ½, ½; 2(c): 0, y, ; 0, , ; 2(d): ½, y, ; ½, _,

       Figure 1.65 Orthorhombic space group F222 (No 22); coordinates of equivalent positions 16(k)

      Special positions with point symmetry 222, 4(a): only one position is given as the other three are generated by the face centring; 0, 0, 0; 4(b): 0, 0, ½; 4(c): , , ; 4(d): , , ; special positions with point symmetry 2, 8(e): x, 0, 0; _, 0, 0; 8(f): 0, y, 0; 0, , 0; 8(g): 0, 0, z; 0, 0, ; 8(h): , , z; , , ½ − z; 8(i): , y, ; , ½ − y, ; 8(j): x, , ; ½ − x, , _.

      There are sixteen general equivalent positions that fall into four groups related by the face‐centring condition. The four sets are related as (0, 0, 0); (½, ½, 0); (½, 0, ½) and (0, ½, ½). Thus, position l, (x, y, z), is related to positions 2–4: (x + ½, у + ½, z); (x + ½, y, z + ½) and (x, y + ½, z + ½). Generation of the remaining equivalent positions by the action of the 2‐fold axes should be straightforward.

       1.18.5.6 Tetragonal I41

The principal axis in the space group I4₁, Fig. 1.66, is a 4₁ screw axis parallel to z.