Anthony R. West - Solid State Chemistry and its Applications

The coordinates of the general and special positions for each space group are listed in International Tables for X‐ray Crystallography together with additional information, as shown in Table 1.30 for P . The general positions are listed first followed by the various special positions; for each, the first column gives the multiplicity , i.e. the number of equivalent positions. The second column is the so‐called Wyckoff notation in inverse alphabetical sequence; the logic behind this labelling scheme is that those positions at the bottom of the list have highest point symmetry and the symmetry may decrease on moving upwards through the list. The third column specifies the point symmetry of the positions. In this case, all the special positions coincide with a centre of symmetry whereas the general position, labelled as 2(i), has no point symmetry.

By common convention, the unique 2‐fold axis in a monoclinic unit cell is labelled b . Unfortunately, this is different to the use of с as the unique axis in tetragonal, trigonal and hexagonal cells but this usage for monoclinic cells is now so well established that it is unlikely to be altered. With b as the unique axis, the unit cell projects onto the xy plane as a rectangle (because γ = 90°), as shown in Fig. 1.62. Since β ≠ 90°, the z axis is not perpendicular to the plane of the paper but is inclined to the vertical.

Table 1.30 Coordinates and labelling of equivalent positions in space group P

Number of positions	Wyckoff notation	Point symmetry	Coordinates of equivalent positions
2	I	1	xyz,
1	h		½, ½, ½
1	g		0, ½, ½
1	f		½, 0, ½
1	e		½, ½, 0
1	d		½, 0, 0
1	c		0, ½, 0
1	b		0, 0, ½
1	a		0, 0, 0

The C ‐centring in space group C 2 means that if the Bravais lattice has a lattice point at the origin (with coordinates 0, 0, 0), it also has an equivalent lattice point in the middle of the side bounded by a and b , at ½, ½, 0. For any position x, y, z in this space group, there will, therefore, be an equivalent position at x + ½, y + ½, z . This C ‐centring has no representation in the right‐hand diagram of Fig. 1.62 but can be seen in the left‐hand diagram; for example, positions 1 and 2 are related by the C ‐centring.

The main symmetry element present in space group C 2 is a 2‐fold rotation axis parallel to b ; it passes through the origin and is coincident with the y axis of the unit cell. The symbol for a 2‐fold rotation axis in the plane of the paper is an arrow. In this case, it is parallel to and coincident with y , passes through the origin x = 0, z = 0, and is shown by arrow d in the right‐hand diagram. The effect of the 2‐fold rotation axis on position 1, left‐hand diagram with coordinates x, y, z, is to generate the equivalent position shown as 3′, with coordinates −x, y, −z .

Other symmetry elements are generated automatically by a combination of this 2‐fold rotation axis and the C ‐centring. Thus, another 2‐fold rotation axis, e , parallel to b, is created which cuts the x axis at ½ and the z axis at 0; positions 1 and 3 are related by this 2‐fold axis, as are positions 3′ and 1‴, positions 4 and 2, etc.

We also find that 2‐fold screw axes have been created automatically in this space group. A 2‐fold screw axis, symbol 2 1, involves a rotation component of 180 oand a translation by ½ in the direction of the screw axis. Two 2 1screw axes, f and g, are shown which are parallel to b , cut the x axis at and are in the plane of the unit cell projection at z = 0. Screw axes in the plane of the paper are represented as half‐arrows. Screw axis f relates positions 1, 4 and 1′ on an imaginary spiral that passes through the unit cell. The 2 1screw axis is a combined translation and rotation operation. Thus, position 1 is translated halfway along y, retaining its x and z values, to the position shown as the dashed circle and then rotated by 180° about the axis parallel to y and at x = , z = 0, to arrive at position 4; it is important to recognise that the dashed circle is not an equivalent position but is drawn merely to show that two operations are involved in the screw axis. Positions 3′, 2, and 3″ are similarly related by the same screw axis.

Space group C 2 has four equivalent positions which are generated by a combination of the C ‐centring and a 2‐fold rotation axis. Starting from position 1, the effect of C ‐centring is to create position 2 which is displaced by (½, ½, 0) from position 1. The effect of the 2‐fold rotation axis, d is to rotate position 1 about the b edge by 180° and create position 3′. As position 1 has a positive z coordinate, 3′ must have a corresponding negative z value. The position equivalent to 3′ that lies inside the unit cell is found by translating to adjacent unit cells in both x and z directions to arrive at a position that is above 3 and is inside the unit cell, i.e. 3 has coordinates 1 − x, y, −z and therefore the equivalent position inside the unit cell is at , y , .