Anthony R. West - Solid State Chemistry and its Applications

The space group C 2 /m contains eight general equivalent positions, all of which may be generated from position 1 by the combined action of the C ‐centring, one 2‐fold axis, and mirror plane. Thus, the C ‐centring creates an equivalent position, 2, after translation by ½, ½, 0. The action of the 2‐fold axis passing through the origin generates 6′ from 1. Position 3 is similarly related to 2 by the action of the 2‐fold axis passing through a = ½, с = 0. Alternatively, 3 may be generated from 6′ by the C ‐centring condition. The mirror plane at b = 0 generates positions 8″ from 1 and 7‴ from 6′. Note that 8″ and 1 are at the same positive с value and that 8″ contains a comma to indicate its enantiomorphic relation to 1. Positions 4 and 5 are related to 3 and 2 by the mirror plane that cuts b at ½; alternatively, 4 and 5 are generated from 7‴ and 8″ by the C ‐centring.

The coordinates of the eight equivalent positions within the cell, together with their number, if shown, are: x, y, z (1); x + ½, y + ½, z (2); ½ − x , ½ + y , ; ½ − x , ½ − y , ; ½ + x , ½ − y , z (5); , y , ; , , ; x , , z (8). These eight positions may be grouped into two sets of four positions that are related by the C ‐centring. The coordinates of both sets are given in Fig. 1.63. Several sets of special positions are possible in this space group, e.g. if y = 0, a 4‐fold set occurs: x , 0, z ; , 0, ; x + ½, ½, z ; ½ − x, ½, . If x = 0, y = 0 and z = ½, a 2‐fold set arises: 0, 0, ½ and ½, ½, ½. All the special positions are listed in Fig. 1.63 caption.

The combination of a mirror plane perpendicular to a 2‐fold axis, together with the C ‐centring, leads to the generation of several other symmetry elements. These include 2 1screw axes parallel to b , centres of symmetry and glide planes. For example, the centre of symmetry created at the origin relates positions 1 and 7‴, 6′ and 8″.

The thick dashed line g in the right‐hand diagram indicates a glide plane for which the translation component is a /2 and reflection is across a plane perpendicular to b . Such a glide plane is called ‘an a glide perpendicular to b ’. Thus, position 1 is translated by a /2 to the position shown as the dashed circle; reflection across the plane, g , which cuts b at leads to position 5. Repetition of the process converts 5 into 1′, which is equivalent to the starting position, 1. Similarly, positions 8, 2 and 8‴ are related by the glide plane which cuts b at .

This primitive orthorhombic space group has 2‐fold rotation axes parallel to x and у and a 2 1screw axis parallel to z . The feature of this space group, Fig. 1.64, which makes the generation of the equivalent positions a little difficult to visualise, is that the 2‐fold rotation axes parallel to у occur at а с height of . Consider first the axis parallel to у at a = 0 and c = . The starting position 1 has a small positive z coordinate of + z ; the twofold axis is at z = . Therefore, position 1 is at ( − z ) below the 2‐fold axis. The new position, 2′, formed by rotation about this axis is, therefore, at ( − z ) above the 2‐fold axis, i.e. it has z coordinate + ( − z ) = 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 2 1axis parallel to z , are automatically generated by the combined action of the other two axes and are not independent of them. This 2 1axis 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.

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 2 1axes are also created, e.g. at a = 0, b = , parallel to с and at b = , c = 0, parallel to a .