Solid State Chemistry and its Applications. Anthony R. West

Optical activity is confined to 15 of the 21 non‐centrosymmetric point groups and piezoelectricity to 20 of these. This is of use in, for example, the search for new materials with piezoelectric activity; it is a waste of time trying to detect piezoelectricity in crystals whose point group is not among the 20 active groups! Crystallographers also make some use of the piezoelectric effect in structure determination. It is a considerable help in solving an unknown structure to know the space group at the outset. If a test for piezoelectricity is carried out with positive results, this limits the choice of space group to the non‐centrosymmetric ones. The absence of piezoelectricity does not necessarily mean, however, that the point group and space group are centrosymmetric.

      1.18.5 Space groups

      The combination of the 32 possible point groups and the 14 Bravais lattices (which in turn are combinations of the 7 crystal systems, or unit cell shapes, and the different possible lattice types) gives rise to 230 possible space groups. All crystalline materials, apart from those showing either quasisymmetry, Section 1.2.2, or those that possess a superstructure with a different, incommensurate periodicity to that of the sublattice, have a structure which belongs to one of these space groups. This does not, of course, mean that only 230 different crystal structures are possible. For the same reason, the human body (from its external appearance) is not the only object to belong to point group teapots also do!

      Space groups are formed by adding elements of translation to the point groups. The space symmetry elements – screw axes and glide planes – are derived from their respective point symmetry elements – rotation axes and mirror planes – by adding a translation step in between each operation of rotation or reflection, Section 1.2.5. A complete tabulation of all possible screw axes and glide planes and their symbols is not given here; instead, symbols are explained as they arise. We also discuss only a few of the simpler space groups; however, once the basic rules have been learned, by working through these examples, there should be no difficulty in understanding and using any space group. The interested reader is recommended to consult the authoritative International Tables for X‐ray Crystallography, Vol 1 (preferably an early edition as later additions contain extra material of relevance only to specialist crystallographers).

      The written symbol of a space group contains between two and four characters. The first character is always a capital letter which corresponds to the lattice type: P, I, A, etc. The remaining characters correspond to some of the symmetry elements that are present. If the crystal system has a unique or principal axis (e.g. the 4‐fold axis in tetragonal crystals), the symbol for this axis appears immediately after the lattice symbol. For the remaining characters, there appear to be different rules for different crystal systems. As these rules are not essential to an understanding of space groups, they are not repeated here.

      Space groups are usually drawn as parallelograms to represent the unit cell, with the plane of the parallelogram corresponding to the xy 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 . These are represented by large open circles; a positive sign indicates that they are above the plane of the paper and a negative sign below. In the latter case, that position will lie in the unit cell below the one that has been defined. A comma inside the equivalent position symbol indicates an enantiomorphic relationship to a second, centrosymmetrically related equivalent position.

      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 (number 2); coordinates of equivalent positions: x, y, z; .

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 , _,