Having thus settled the directions of the crystallographic axes and their lengths, it is the next step which reveals the remarkable law to which reference was made at the opening of this chapter. For we find that all other faces on the crystal, however complicated and rich in faces it may be, cut off lengths from the axes which are represented by low whole numbers, that is, either 2, 3, 4, or possibly 5, and very rarely more than 6 unit lengths. By far the greater number of faces do not cut off more than three unit lengths from any axis. Prof. Miller of Cambridge, in the year 1839, gave us a most valuable mode of labelling and distinguishing the various faces by a symbol involving these three values, employed, however, not directly but in an indirect yet very simple manner. If m, n, r be the three numbers expressing the intercepts cut off by a face on the three axes, a, b, c respectively, and if the Millerian index numbers be represented by h, k, l, then—
| m = a h, | n = b k, | r = c l, | |
| or, | h = a m, | k = b n, | l = c r. |
Each figure or “index” of the Millerian symbol is thus inversely proportional to the length of the intercept on the axis concerned. The intercepts themselves are used as symbols in another mode of labelling crystal faces, suggested by Weiss, but this method proves too cumbersome in practice.
The Millerian symbol of a face is always placed within ordinary curved brackets ( ), but if the symbol is to stand for the whole set of faces composing the form, the brackets are of the type { }. Thus the Millerian symbol of the fourth face (that in the top-right front octant), determinative of the unit axial lengths, is (111), as shown in Fig. 46, the face in question being marked with this symbol; while the symbol {111} indicates the set of faces of the whole or such part of the double pyramid as composes the unit form. In the triclinic system this form only consists of the face (111) and the parallel one (̄1̄1̄1), but in the case of the regular octahedron of the cubic system it embraces all the eight faces. The triclinic octahedron, Fig. 46, is thus made up of four forms of two faces each. A negative sign over an index indicates interception on the axis a behind the centre, on the axis b to the left of the centre, or on the vertical axis c below the centre.
To take an actual example, suppose a face other than the primary one to make the intercepts on the axes 4, 2, 1; in this case h = a/4, k = b/2, and l = c/1, that is, when referred to the fundamental primary form for which a, b, c are each unity, h = ¼, k = ½, l = 1, or, bringing them to whole numbers by multiplying by 4, h = 1, k = 2, c = 4, and the symbol in Millerian notation is (124). Again, suppose we wish to find the intercepts on the three cubic axes made by the face (321) of the hexakis octahedron shown in Fig. 21. To get each intercept we multiply together the two other Millerian indices, and if necessary afterwards reduce the three figures obtained to their simplest relative values. For the face (321) we obtain 2, 3, 6. This means that the face (321) in the top-right-front octant of the hexakis octahedron cuts off two unit lengths of axis a, three unit lengths of axis b, and six unit lengths of axis c. No fractional parts thus ever enter into the relations of the axial lengths intercepted by any face on a crystal, and the whole numbers representing these relations are always small, the number 6 being the usual limit.
This important law is known as the “Law of Rational Indices,” and is the corner-stone of crystallography. A forecast of it was given in Chapter III., in describing how it was first discovered by Haüy, and it was shown how impressed Haüy was with its obvious significance as an indication of the brick-like nature of the crystal structure. What the “bricks” were, Haüy was not in a position to ascertain with certainty, as chemistry was in its infancy, and Dalton’s atomic theory had not then been proposed.
That Haüy had a shrewd idea, however, that the structural units were the chemical molecules, and that while the main lines of symmetry were determined by the arrangement of the molecules its details were settled by the arrangement of the atoms in the molecules, is clear to any one who reads his 1784 “Essai” and 1801 “Traité,” and interprets his molécules intégrantes and élémentaires in the light of our knowledge of to-day.
Before we pass on, however, to consider the modern development of the real meaning of the law of rational indices, as revealed by recent work on the internal structure of crystals, it will be well to consider first, in the next chapter, a few more essential facts as to crystal symmetry, and the current mode of constructing a comprehensive, yet simple, plan of the faces present on a crystal.
CHAPTER VI
THE DISTRIBUTION OF CRYSTAL FACES IN ZONES, AND THE MODE OF CONSTRUCTING A PLAN OF THE FACES.
It will have been clear from the facts related in the previous chapters that the salient property possessed by all crystals, when ideal development is permitted by the circumstances of their growth, and the substance is not one of unusual softness or liable to ready distortion, is that the exterior form consists of and is defined by truly plane faces inclined to each other at angles which are specific and characteristic for each definite chemical substance; and that these angles are in accordance with the symmetry of some particular one of the thirty-two classes of crystals, and are such as cause the indices of the faces concerned to be rational small numbers.