Читать книгу Solid State Chemistry and its Applications - Anthony R. West - Страница 85

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 C2 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 C2 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 2₁, involves a rotation component of 180^o and a translation by ½ in the direction of the screw axis. Two 2₁ screw 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 2₁ screw 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 C2 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, .

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 C2. All the other positions shown in Fig. 1.62 are created by translation to adjacent unit cells; the other 2‐fold rotation and 2₁ screw 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 C2 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).