Читать книгу Solid State Chemistry and its Applications - Anthony R. West - Страница 83

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.