(d) Zones.
An important consequence of the law of rational intercepts is the arrangement of the faces of a crystal in zones. All faces, whether they belong to one or more simple forms, which intersect in parallel edges are said to lie in the same zone. A line drawn through the centre O of the crystal parallel to these edges is called a zone-axis, and a plane perpendicular to this axis is called a zone-plane. On a cube, for example, there are three zones each containing four faces, the zone-axes being coincident with the three tetrad axes of symmetry. In the crystal of zircon (fig. 88) the eight prism-faces a, m, &c. constitute a zone, denoted by [a, m, a′, &c.], with the vertical tetrad axis of symmetry as zone-axis. Again the faces [a, x, p, e′, p′, x″′, a″] lie in another zone, as may be seen by the parallel edges of intersection of the faces in figs. 87 and 88; three other similar zones may be traced on the same crystal.
The direction of the line of intersection (i.e. zone-axis) of any two planes (hkl) and (h1k1l1) is given by the zone-indices [uvw], where u = kl1 − lk1, v = lh1 − hl1, and w = hk1 − kh1, these being obtained from the face-indices by cross multiplication as follows:—
Any other face (h2k2l2) lying in this zone must satisfy the equation
h2u + k2v + l2w = 0.
This important relation connecting the indices of a face lying in a zone with the zone-indices is known as Weiss’s zone-law, having been first enunciated by C. S. Weiss. It may be pointed out that the indices of a face may be arrived at by adding together the indices of faces on either side of it and in the same zone; thus, (311) in fig. 12 lies at the intersections of the three zones [210, 101], [201, 110] and [211, 100], and is obtained by adding together each set of indices.
(e) Projection and Drawing of Crystals.
The shapes and relative sizes of the faces of a crystal being as a rule accidental, depending only on the distance of the faces from the centre of the crystal and not on their angular relations, it is often more convenient to consider only the directions of the normals to the faces. For this purpose projections are drawn, with the aid of which the zonal relations of a crystal are more readily studied and calculations are simplified.
| Fig. 10.—Stereographic Projection of a Cubic Crystal. |