Solid State Chemistry and its Applications. Anthony R. West

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Solid State Chemistry and its Applications - Anthony R. West

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      The space group C2/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, z overbar; ½ − x, ½ − y, z overbar; ½ + x, ½ − y, z (5); x overbar, y, z overbar; x overbar, y overbar, z overbar; x, y overbar, 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; x overbar, 0, z overbar; x + ½, ½, z; ½ − x, ½, z overbar. 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 21 screw 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″.

       1.18.5.4 Orthorhombic P2221

      This primitive orthorhombic space group has 2‐fold rotation axes parallel to x and у and a 21 screw 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 one fourth. Consider first the axis parallel to у at a = 0 and c = one fourth. The starting position 1 has a small positive z coordinate of +z; the twofold axis is at z = one fourth. Therefore, position 1 is at (one fourthz) below the 2‐fold axis. The new position, 2′, formed by rotation about this axis is, therefore, at (one fourthz) above the 2‐fold axis, i.e. it has z coordinate one fourth + (one fourthz) = 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

Schematic illustration of orthorhombic space group P2221 (No 17); Coordinates of equivalent positions 4€.

      Figure 1.64 Orthorhombic space group P2221 (No 17); Coordinates of equivalent positions 4(e): x, y, z; bold-italic x overbar, y, ½ − z; x, bold-italic y overbar 
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