Anthony R. West - Solid State Chemistry and its Applications

Figure 1.62 Monoclinic space group C2 (No 5); coordinates of equivalent positions 4(c): x, y, z; ; x + , y + , z; – x, + y, . Special positions, point symmetry 2, 2(b): 0, ½, z; 2(a): 0, 0, z.

The same rotation axis d creates position 4′ from position 2, whose equivalent position inside the unit cell is above position 4 with coordinates ½ – x , ½ + y , . Hence, in summary, we may say that the C ‐centring creates a second equivalent position and the effect of the 2‐fold axis is to create two more equivalent positions, to give a total of 4 equivalent positions in space group C 2. All the other positions shown in Fig. 1.62 are created by translation to adjacent unit cells; the other 2‐fold rotation and 2 1screw axes are generated automatically and do not generate any extra equivalent positions.

The coordinates of the four positions that lie inside the unit cell can be grouped into two sets: x, y, z ; , y , and x + ½, y + ½, z ; ½ − x , ½ + y , . The second set is related to the first by the lattice centring (i.e. by adding ½, ½, 0 to the coordinates). It is common practice (e.g. in International Tables of X‐ray Crystallography ) to list only those positions that belong to the first set but at the same time specify that other positions are created by the lattice centring. This leads to considerable shortening and simplification in labelling the equivalent positions of the more complex and higher symmetry space groups.

The general positions in space group C 2 are 4‐fold but if they lie on the 2‐fold rotation axes, their number is reduced to two and they become special positions. Thus, if x = z = 0, the two positions have coordinates 0, y , 0 and , y + ½, 0. A second set of special positions arises when x = 0, z = ½ (the reader may like to check that there is a 2‐fold axis parallel to b and at x = 0, z = ½ that is not indicated in Fig. 1.62: it is at height c /2 above axis d ).

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 2 1screw axes parallel to b and intersecting a at and . All of these 2 and 2 1axes 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, .