| OA | : | OB | : | OC | . |
| h | k | l |
Thus for a face parallel to the plane A Be the intercepts are in the ratio OA : OB : Oe, or
| OA | : | OB | : | OC |
| 1 | 1 | 2 |
and for a plane fgC they are Of : Og : OC or
| OA | : | OB | : | OC | . |
| 2 | 3 | 1 |
Now the important relation existing between the faces of a crystal is that the denominators h, k and l are always rational whole numbers, rarely exceeding 6, and usually 0, 1, 2 or 3. Written in the form (hkl), h referring to the axis OX, k to OY, and l to OZ, they are spoken of as the indices (Millerian indices) of the face. Thus of a face parallel to the plane ABC the indices are (111), of ABe they are (112), and of fgC (231). The indices are thus inversely proportional to the intercepts, and the law of rational intercepts is often spoken of as the “law of rational indices.”
The angular position of a face is thus completely fixed by its indices; and knowing the angles between the axial planes and the parametral plane all the angles of a crystal can be calculated when the indices of the faces are known.
| Fig. 9.—Crystallographic axes of reference. |
Although any set of edges formed by the intersection of three planes may be chosen for the crystallographic axes, it is in practice usual to select certain edges related to the symmetry of the crystal, and usually coincident with axes of symmetry; for then the indices will be simpler and all faces of the same simple form will have a similar set of indices. The angles between the axes and the ratio of the lengths of the parameters OA : OB : OC (usually given as a : b : c) are spoken of as the “elements” of a crystal, and are constant for and characteristic of all crystals of the same substance.
The six systems of crystal forms, to be enumerated below, are defined by the relative inclinations of the crystallographic axes and the lengths of the parameters. In the cubic system, for example, the three crystallographic axes are taken parallel to the three tetrad axes of symmetry, i.e. parallel to the edges of the cube (fig. 5) or joining the opposite corners of the octahedron (fig. 3), and they are therefore all at right angles; the parametral plane (111) is a face of the octahedron, and the parameters are all of equal length. The indices of the eight faces of the octahedron will then be (111), (111), (111), (111), (111), (111), (111), (111). The symbol {111} indicates all the faces belonging to this simple form. The indices of the six faces of the cube are (100), (010), (001), (100), (010), (001); here each face is parallel to two axes, i.e. intercepts them at infinity, so that the corresponding indices are zero.