Crystals

by A. E. H. Tutton

It will also be clear that, given the presence of any face other than the three axial planes, the symmetry of the class—supposing the crystal to exhibit some development of symmetry and not to belong to class 32, the general case possessing no symmetry—will require the repetition of this face a definite number of times on other parts of the crystal. Such a set of faces possessing the same symmetry value we have already learnt to call a “Form,” and the faces composing it will have the same Millerian index numbers in their symbols, but differently arranged and with negative signs over those which relate to the interception of the back part of the a axis, the left part of the b axis, or the lower part of the vertical c axis; that is, parts to the front and right, and above, the centre of intersection of the three crystal axes are considered as the positive parts of those axes.

A form, if of general character, that is, if composed of faces each of which is inclined to all three axes, will comprise more faces the higher the symmetry. Thus, in the cubic system, the form shown in Fig. 21, the hexakis octahedron, comprises as many as forty-eight faces, all covered by the form symbol {321}; while in the rhombic system the highest number of faces in a form is eight, in the monoclinic only four, and in the triclinic system two. It will also have become clear that the law of rational indices limits the number of forms possible of any one type. For instance, very few hexakis octahedra are known, the most frequently occurring ones besides {321} being {421}, {531}, and {543}. Forms, of any class, possessing higher indices than these are very rare, especially in the systems of lower symmetry.

Fig. 47.—The Spherical Projection.

We next come to a further very interesting fact about crystals. Let us imagine a crystal, on which the faces are fairly evenly developed, to be placed in the middle of a sphere of jelly, as indicated in Fig. 47 (reproduced from a Memoir by the late Prof. Penfield), so that the centre or origin of the axial system of the crystal and the centre of the sphere coincide. Let us now further imagine that long needles are stuck through the jelly and the crystal, one perpendicular to each crystal face, and so as to reach the centre. The crystal represented in Fig. 47 is a combination of the cube a, octahedron o, and rhombic dodecahedron d. If such a thing as we have imagined were possible, we should find that the needles would emerge at the surface of the sphere in points which would lie on great circles, that is, on circles which represent the intersection of the sphere by planes passing through the centre. Moreover, the points would be distributed along these circles at regularly recurring angular positions, corresponding to the symmetry of the crystal. If the crystal belonged to one of the higher systems of symmetry, it would happen that four of the points on at least one of these great circles, and possibly on three of them, would be 90° apart, that is, would be at the ends of rectangular diameters, which would most likely be the axes of reference. The other points would be distributed symmetrically on each side of these four points.

The great circles on which the points are thus symmetrically distributed—and they may legitimately be taken to represent the faces, for tangent planes to the sphere at these points would be parallel to the faces—are known as “zone circles,” and the faces represented by the points on any one of them form a “zone.” Now a zone of faces has this practical property, that when the crystal is supported so as to be rotatable about the zone axis—which is parallel to the edges of intersection of all the faces composing the zone, and is the normal to the plane of the great circle representing the zone—and a telescope is directed towards the crystal perpendicularly to the zone axis, while a bright object such as an illuminated slit is arranged conveniently so as to be reflected from any face of the crystal into the telescope, an image of it being thus visible in the latter, then it will be found that on rotating the crystal a similar image will be seen reflected in the telescope from every face of the zone in turn. Moreover, when the crystal is mounted on a graduated circle, the angle of rotation between the positions of adjustment to the cross-wires of the telescope of any two successive images, reflected from adjacent faces of the crystal, is actually the angle between the two points representing the faces concerned on the zone circle, and is the supplement of the internal dihedral angle between the two crystal faces themselves. It is, in fact, the angle between the normals (perpendiculars) to the two faces, the angle which is measured on the goniometer.

This is, indeed, the very simple principle of the reflecting goniometer, invented by Wollaston in the year 1809, and which in its modern improved form is the all-important principal instrument of the crystallographer’s laboratory. The work with it consists largely in the measurement of the angles between the faces in all the principal zones developed on the crystal. The very fact, however, that crystal faces occur so absolutely accurately in zones immeasurably lightens the labours of the crystallographer, and is one of prime importance.

Fig. 48.—The Reflecting Goniometer.

The most accurate and convenient modern form of reflecting goniometer, reading to half-minutes of arc, and provided with a delicate adjusting apparatus for the crystal, is shown in Fig. 48. It is constructed by Fuess of Berlin.