Crystallographic and Geometrical Symmetry
Crystallographic symmetry must not be confused with geometrical symmetry. Crystallographic symmetry depends upon the internal atomic structure of the crystal, and as the arrangement of the atoms is the same for parallel planes, it follows that the angular position is the only factor concerned, and that the sizes of like faces and their distances from a plane or centre of symmetry are of no importance in this connection. This is illustrated in Fig. 13, which shows a regular octahedron with faces equally developed and a distorted octahedron with some faces larger than others. If such a distorted octahedron was examined with a goniometer it would be found that the interfacial angles were exactly the same as in the regular octahedron.
Crystals in which like faces are equally developed and are equal distances from the centre of the crystal are rare; but for convenience of study and of representation by diagrams, it is necessary to deal with crystals in their simplest and due to some restraint on directions or to a greater supply of material being available in one direction as compared with another, most intelligible form, and that is when they have perfect geometrical symmetry.
Most crystals occur in distorted forms, having like faces not of the same size and not in the same geometrically symmetrical position. In many cases of distorted crystals the crystallographic symmetry has been made out from the fact that like faces have like properties. Etch-marks produced by solvents acting on the crystal faces, the behaviour towards heat and electricity, the hardness, luster, etc., of the faces, have revealed the true symmetry of the distorted crystals. This is illustrated in the quartz crystal shown in Fig. 14, where the etch-marks are similar on like faces.
Distortion in crystals may be growth of the crystal in certain directions or to a greater supply of material being available in one direction as compared with another.
The term habit is used to denote the characteristic shapes of crystals arising from variations in the number, size and shape of the faces; the distorted octahedron shown in Fig. 13 has a tabular habit; in Fig. 15 are shown two habits of apophyllite crystals.
Crystallographic Axes
In solid geometry the position of a plane in space is given by the intercepts (or the lengths cutoff) that the plane makes on three given lines called axes. This method of treatment is employed in crystallography, and the axes are termed the crystallographic axes. Whenever there is present a suitable number of axes of symmetry they are chosen as the crystallographic axes. The crystallographic axes intersect at the origin.
Parameters
The parameters of a crystal face are the ratios of the distances from the origin at which the face cuts the crystallographic axes, - that is, the parameters are the ratios of the intercepts. In Fig. 16, OX, OY, OZ, represent the crystallographic axes, and ABC is a crystal face making intercepts of OA on OX, OB on OY, and OC on OZ. The parameters of the face ABC are given by the ratio of OA, OB, and OC. It is convenient to take the relative intercepts of this face as standard lengths for the purpose of representing the position of any other face, such as DEF. In the case of the face DEF, OD is equal to OA, OE is twice OB, and OF is half OC, and therefore 1/1, 2/1, 1/2 are the parameters of DEF with reference to the standard face ABC.
The form whose face is taken as intersecting the axes at the unit lengths which are to be used for measuring the intercepts made by other forms on the same axes is called the fundamental, parametral or unit form. The selection of a suitable unit form depends on the properties and nature of the crystals. A form well developed, or commonly occurring, or parallel to which there is a good cleavage, is usually selected for this purpose.
The parameters of the unit form can be obtained by measurement, and can be expressed as multiples of one of their number. Take, for example, gypsum. It is found that the most commonly occurring form in gypsum crystals which makes intercepts with all three crystallographic axes does so in the ratio of 0′6899:1 :0,4124. This expression is called the axial ratio, and simply means that the standard or unit form cuts one axis at a distance represented by 0′6899, the second axis at a distance represented by 1, and the third axis at a distance represented by 0′4124. When we use this unit form to measure the intercepts, or to obtain the parameters, of any other form that cuts all three axes we shall do so by taking 0•6899 as our unit of measurement along the first axis, 1 along the second axis, and 0′4124 along the third axis.