Читать книгу Solid State Chemistry and its Applications - Anthony R. West - Страница 86

This space group, shown in Fig. 1.63, is also C‐centred and has, as its principal symmetry elements, a mirror plane perpendicular (/) to a 2‐fold axis. The 2‐fold axis is parallel to b, by convention, and therefore the mirror plane is the xz plane. Two mirror planes are present in the cell; they intersect b at 0 and ½ and are shown as thick vertical lines in Fig. 1.63. As in the space group C2 , there are two 2‐fold rotation axes, parallel to b and intersecting a at 0 and ½, and two 2₁ screw axes parallel to b and intersecting a at and . All of these 2 and 2₁ axes are at с height equal to zero. An additional set of axes, not shown, occurs at с = ½. Also present, as discussed later, are centres of symmetry and glide planes.

Figure 1.63 Monoclinic space group C2/m (No 12). Coordinates of general equivalent positions 8(j): x, y, z; x, , z; , and ½ + x, ½ + y, z; ½ + x, ½ – y, z; ½ – x, ½ + y, ; ½ – x, ½ – y, . Special positions with point symmetry 2/m, 2(a): 0, 0, 0; ½, ½, 0; 2(b): 0, ½, 0; ½, 0, 0; 2(c): 0, 0, ½; ½, ½, ½; 2(d):0, ½, ½; ½, 0, ½; special positions with point symmetry , 4(e): , , 0; , , 0; , , 0; , , 0; 4(f): , , ½; , , ½; , , ½; , , ½; special positions with point symmetry 2, 4(g): 0, y, 0; 0, , 0; ½, ½ + y, 0; ½, ½ − y, 0; 4(h): 0, y], ½; 0, , ½; ½, ½ + y, ½; ½, ½ − y, ½; special position with point symmetry m, 4(i): x, 0, z; , 0, ; ½ + x, ½, z; ½ − x ½, .

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, ; 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 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″.

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 .