COMPOSITION . Most chemical elements and compounds are capable of assuming the crystalline See also:

• CONDITION (Lat. condicio, from condicere, to agree upon, arrange; not connected with conditio, from condere, conditum, to put together)

He further pointed out that the crystals must have grown in a liquid by the addition of layers of material upon the faces of a See also:

• NUCLEUS (Lat. for the kernal of a nut, nux, the stone of fruit)

J. B. L. de Rome de l'Isle (Essai de cristallographie, See also:

• PARIS
• PARIS (also called ALEXANDROS)
• PARIS, ALEXIS PAULIN (1800-x881)
• PARIS, BRUNO PAULIN GASTON (1839-1903)
• PARIS, FRANCOIS DE (169o-1727)
• PARIS, LOUIS PHILIPPE ALBERT
• PARIS, TREATIES OF (1814-1815)

I shows a scalenohedron ,(vkaXrlvbs; uneven) built up in this manner of rhombohedra; and fig. 2 a regular octahedron builtup of cubic elements, such as are given by the cleavage of See also:

• GALENA

These two systems (the See also:

• MONOCLINIC

Optically biaxial crystals were afterwards shown by J. F. W. See also:

• HERSCHEL, CAROLINE LUCRETIA (1750-1848)
• HERSCHEL, SIR F
• HERSCHEL, SIR FREDERICK WILLIAM (1738-1822)
• HERSCHEL, SIR JOHN FREDERICK WILLIAM, BART

'jµ1-, See also:

• HALF

See also:

• MARX, HEINRICH KARL (1818-1883)

3. Law of Rational Intercepts or Indices. According to the first law, the angles between corresponding faces of all crystals of the same chemical substance are always the same and are characteristic of the substance. (a) Symmetry of Crystals. Crystals may, or may not, be symmetrical with respect to a point, a See also:

• LINE

0: by rotating the crystal about this axis through a right angle (go°) it reaches a position such that the See also:

• ORIENTATION

About this axis there may be rotation of 18o°, and only twice in a complete revolution of 36o° (=18o°X 2) is the crystal brought into interchangeable positions. There being six pairs of parallel edges on an octahedron; there are consequently six dyad axes of symmetry. A regular octahedron thus possesses thirteen axes of symmetry (of three kinds), and there are the same number in the cube. Fig. 5 shows the three tetrad (or tetragonal) axes (aa), four triad (or trigonal) axes (pp), and six dyad (diad or See also:

• DIAGONAL (Gr. &a, through, ywvia, a corner)

Corresponding planes on either side of a plane of symmetry are inclined to it at equal angles. The octahedron can also be divided by similar planes of symmetry passing through the corners ala2a,az and a2a3a2a3• These three similar planes of symmetry are called the cubic planes of symmetry, since they are parallel to the faces of the cube (compare figs. 6-8, showing combinations of the p octahedron and the cube). A regular octahedron can also be divided symmetrically into two equal and similar portions by a plane passing through the corners a3 and a3, the middle points d of the edges alai and d1a2, and the centre 0 (fig. 4). This is called a dodecahedral plane of symmetry, being parallel to the face of the rhombic See also:

• DODECAHEDRON (Gr. Melia, twelve, and ESpa, a face or base)

A single face ala2a3 (figs. 3 and 4) may be repeated by certain of the elements of symmetry to give the whole eight faces of the octahedron. Thus, by rotation about the vertical, tetrad axis aaa3' the four upper faces are obtained; and by rotation of these about one or other of the See also:

• HORIZONTAL

6 the corners of the cube are symmetrically replaced or truncated by the faces of the octahedron, and in fig. 8 those of the octahedron are truncated by the cube. (c) Law of Rational Intercepts. For axes of reference, OX, OY, OZ (fig. 9), take any three edges formed by the intersection of three faces of a crystal. These axes are called the crystallographic axes, and the planes in which they See also:

• LIE, JONAS LAURITZ EDEMIL (1833—1908)
• LIE, MARIUS SOPHUS (1842–1899)

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. 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 indiceg will be simpler and all faces of the same simple form will have a similar set of Z indices. The angles between FIG. 9.-Crystallographic axes of the axes and the ratio of the reference. 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 (III) 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 (III), (Iii), (See also:

• ILI

(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 See also:

• ZONE (Gr. 'WV77, a girdle, from 'wvvLvay to gird)

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, Io1], [201, IIO] and [211, See also:

• IOO

In fig. Io is shown the stereographic projection, or stereogram, of a573 cubic crystal; al, a2, &c. are the poles of the faces of the cube. ol, 02, &c. those of the octahedron, and d', d2, &c. those of the rhombic dodecahedron. The straight lines and circular arcs are the projections on the equatorial plane of the great circles in which the nine planes of symmetry intersect the sphere. A drawing of a crystal showing a combination of the cube, octahedron and rhombic dodecahedron is shown in fig. II, in which the faces are lettered the same as the corresponding poles in the projection. From the zone-circles in the projection and the parallel edges in the drawing the zonal relations of the faces are readily seen: thus [alold5], [aldla5], [a5old2l, &c. are zones. A stereographic projection of a See also:

• RHOMBOHEDRAL

In this Cubic Crystal. case all zones are represented by straight lines. Fig. 12 is the gnomonic projection of a cubic crystal, the plane of projection being tangent to the sphere at the pole of an octahedral face (III), which is therefore in the centre of the projection. The indices of the several poles are given in the figure. In drawing crystals the simple plans and elevations of descriptive See also:

• GEOMETRY

II) a cube face a5 is first placed parallel to the plane on which the crystal is to be projected and with one set of edges vertical; the crystal is then turned through a small angle about a vertical axis until a second cube face a2 comes into view, follows: and the eye is then raised so that a third cube face a' may be seen. (f) Crystal Systems and Classes. According to the mutual inclinations of the crystallographic axes of reference and the lengths intercepted on them by the parametral plane, all crystals fall into one or other of six groups or systems, in each of which there are several classes depending on the degree of symmetry. In the brief description which follows of these six systems and thirty-two. classes of crystals we shall proceed from those in which the symmetry is most complex to those in which it is simplest. 1. CUBIC SYSTEM (Isometric ; Regular ; Octahedral ; Tesseral). In this system the three crystallographic axes of reference are all at right angles to each other and are equal in length. They are parallel to the edges of the cube, and in the different classes coincide either with tetrad or dyad axes of symmetry. Five classes are included in this system, in all of which there are, besides other elements of symmetry, four triad axes.