The structure of α and β quartz

Abstract

The quartz crystal has a trigonal axis and three digonal axes, which are at right angles to the trigonal axis. It has no other element of symmetry. It belongs to Class 18 of the 32 classes into which crystals may be divided by their outside appearance. It was shown in 1913, by the methods of X-ray analysis, which were then new, that the unit cell contains three molecules of SiO 2 , which are so arranged that a revolution of 120° round the principal axis, coupled with a translation c /3 along it, brings each molecule into the exact position previously occupied by one of its companions. The trigonal axis is in fact a screw axis. The value of c is 5·375 A. U. The distance between an axis and each of its six equidistant neighbours is 4·89 A. U. Each silicon molecule possesses a digonal axis. This particular arrangement of molecules is known in mathematical crystallography as that of the space-group D 4 3 or D 6 3 , according to the rotatory sense of the lattice. At this stage of the work four parameters, still remained to be determined before the positions of the atoms in the crystal structure could be defined. The position will be clear from a consideration of fig. 1, which is reproduced from ‘X-rays and Crystal Structure,’ p. 261. Each of the two diagrams in the figure shows the relation between three molecules, forming the content of the unit cell, and derivable from each other in the manner described. In one, the screw axis is in the plane of the paper, in the other at right angles to it. The digonal axes are also shown. The distance of the silicon (black disc) from the axis is one of the unknowns: the spatial relations of the oxygens to the silicons require for their definition three more. The diagram of fig. 2 shows various possible positions of the silicon atoms when projected upon the basal plane. They must lie on certain lines, as shown, in order to satisfy the symmetry conditions.