plane of the unit cell. By convention, Fig. 1.60(a), the origin is taken as the top left‐hand corner, with y horizontal, x downwards and the positive z direction out of the plane of the paper and pointing towards the reader. For each space group, two parallelograms are used, the left‐hand one gives the equivalent positions; the right‐hand one gives the symmetry elements. Let us see some examples. Each one introduces at least one new feature.
1.18.5.1 Triclinic P
This space group is primitive and centrosymmetric; it is shown in Fig. 1.60(b, c). The right‐hand diagram shows the symmetry elements: there are centres of symmetry (shown as small open circles) at the origin (t), midway along the a and b edges and in the middle of the С face (i.e. the face bounded by a and b). Additional centres of symmetry, not shown, occur in the middle of the other faces, halfway along the с edge and at the body centre of the unit cell.
The left‐hand diagram gives the equivalent positions in space group
To find the equivalent positions in the space group, it is necessary, as with point groups, to choose a starting position and operate on this position with the various symmetry elements that are present. The conventional starting position is at 1, close to the origin and with small positive values of x, y and z. This position must be present in all other unit cells (definition of the unit cell); three are shown as 1′, 1″ and 1‴.
Figure 1.60 (a) The convention used to label axes and origin of space groups. (b, c) Triclinic space group P
Consider now the effect of the centre of symmetry, t, at the origin of the unit cell. This acts upon position 1 to create position 2. The minus sign at 2 indicates a negative z height and the comma shows its enantiomorphic relation to position 1. Positions 2′, 2″ and 2‴ in Fig. 1.60 are automatically generated from position 2 by translation because they are equivalent positions in adjacent cells.
To understand the meaning of an enantiomorphic relationship, the effect of an inversion operation is to convert a left‐handed object into a right‐handed one and vice versa. This is illustrated in Fig. 1.61 for two tetrahedra that are positioned so as to be related to each other by inversion through a centre of symmetry. Individual tetrahedra do not possess a centre of symmetry, whereas groupings of tetrahedra may possess one, such as shown in Fig. 1.61. In addition, if the tetrahedra themselves are chiral, such as the molecule CHFBrI with the four different corners represented by 1, 2, 3, 4 in Fig. 1.61, then the centrosymmetric partner in the configuration shown is a different isomer with the corner arrangement 1′, 2′, 3′, 4′.
The next step is to write down the coordinates of the equivalent positions in the unit cell. This is done in the form x, y, z where x, y and z are the fractional distances, relative to the unit cell edge dimensions, from the origin of the cell. Let position 1, Fig. 1.60, have fractional coordinates x, y, z; positions 1′, 1″ and 1‴ in adjacent unit cells are given by adding 1 to the relevant coordinates i.e. x, 1 + y, z for 1′, 1 + x, 1 + y, z for 1″ and 1 + x, y, z for 1‴. Position 2 is the centrosymmetric partner position of 1, i.e. − x, −y, −z. Position 2″ is then in the next unit cell at⋯1 – x, 1 – y, –z, etc. Thus, if a position lies outside the unit cell under consideration, an equivalent position within the unit cell can be found, usually by adding or subtracting 1 from one or more of the fractional coordinates. Position 2″ is outside the cell because it has a negative z value; the equivalent position inside the cell is given by a displacement of one unit cell length in the z direction to give coordinates 1 – x, l – y, 1 – z. These coordinates are written in shorthand as
Figure 1.61 Two tetrahedra in a centrosymmetric arrangement.
In summary, therefore, the unit cell in space group P
Although only one centre of symmetry is necessary to generate the equivalent positions in P
The positions x, y, z and
The coordinates of the general and special positions for each space group are
