interpenetrating regular point-systems, each of which is formed
of similar atoms; each of these point-systems is built up from
a number of interpenetrating space-lattices, each of the latter
being formed from similar atoms occupying parallel positions.
All the space-lattices of the combined system are geometrically
identical, or are characterized by the same elementary parallelopipedon.”
A complete résumé, with references to the literature, will be found in “Report on the Development of the Geometrical Theories of Crystal Structure, 1666–1901” (British Assoc. Rep., 1901).
II. PHYSICAL PROPERTIES OF CRYSTALS.
Many of the physical properties of crystals vary with the direction in the material, but are the same in certain directions; these directions obeying the same laws of symmetry as do the faces on the exterior of the crystal. The symmetry of the internal structure of crystals is thus the same as the symmetry of their external form.
(a) Elasticity and Cohesion.
The elastic constants of crystals are determined by similar methods to those employed with amorphous substances, only the bars and plates experimented upon must be cut from the crystal with known orientations. The “elasticity surface” expressing the coefficients in various directions within the crystal has a configuration symmetrical with respect to the same planes and axes of symmetry as the crystal itself. In calcite, for instance, the figure has roughly the shape of a rounded rhombohedron with depressed faces and is symmetrical about three vertical planes. In the case of homogeneous elastic deformation, produced by pressure on all sides, the effect on the crystal is the same as that due to changes of temperature; and the surfaces expressing the compression coefficients in different directions have the same higher degree of symmetry, being either a sphere, spheroid or ellipsoid. When strained beyond the limits of elasticity, crystalline matter may suffer permanent deformation in one or other of two ways, or may be broken along cleavage surfaces or with an irregular fracture. In the case of plastic deformation, e.g. in a crystal of ice, the crystalline particles are displaced but without any change in their orientation. Crystals of some substances (e.g. para-azoxyanisol) have such a high degree of plasticity that they are deformed even by their surface tension, and the crystals take the form of drops of doubly refracting liquid which are known as “liquid crystals.” (See O. Lehmann, Flüssige Kristalle, Leipzig, 1904; F. R. Schenck, Kristallinische Flüssigkeiten und flüssige Krystalle, Leipzig, 1905.)
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| Fig. 91.—Glide-plane of Calcite. |
In the second, and more usual kind of permanent deformation without fracture, the particles glide along certain planes into a new (twinned) position of equilibrium. If a knife blade be pressed into the edge of a cleavage rhombohedron of calcite (at b, fig. 91) the portion abcde of the crystal will take up the position a′b′cde. The obtuse solid angle at a becomes acute (a′), whilst the acute angle at b becomes obtuse (b′); and the new surface a′ce is as bright and smooth as before. This result has been effected by the particles in successive layers gliding or rotating over each other, without separation, along planes parallel to cde. This plane, which truncates the edge of the rhombohedron and has the indices (110), is called a “glide-plane.” The new portion is in twinned position with respect to the rest of the crystal, being a reflection of it across the plane cde, which is therefore a plane of twinning. This secondary twinning is often to be observed as a repeated lamination in the grains of calcite composing a crystalline limestone, or marble, which has been subjected to earth movements. Planes of gliding have been observed in many minerals (pyroxene, corundum, &c.) and their crystals may often be readily broken along these directions, which are thus “planes of parting” or “pseudo-cleavage.” The characteristic transverse striae, invariably present on the cleavage surfaces of stibnite and cyanite are due to secondary twinning along glide-planes, and have resulted from the bending of the crystals.
One of the most important characters of crystals is that of “cleavage”; there being certain plane directions across which the cohesion is a minimum, and along which the crystal may be readily split or cleaved. These directions are always parallel to a possible face on the crystal and usually one prominently developed and with simple indices, it being a face in which the crystal molecules are most closely packed. The directions of cleavage are symmetrically repeated according to the degree of symmetry possessed by the crystal. Thus in the cubic system, crystals of salt and galena cleave in three directions parallel to the faces of the cube {100}, diamond and fluorspar cleave in four directions parallel to the octahedral faces {111}, and blende in six directions parallel to the faces of the rhombic dodecahedron {110}. In crystals of other systems there will be only a single direction of cleavage if this is parallel to the faces of a pinacoid; e.g. the basal pinacoid in tetragonal (as in apophyllite) and hexagonal crystals; or parallel (as in gypsum) or perpendicular (as in mica and cane-sugar) to the plane of symmetry in monoclinic crystals. Calcite cleaves in three directions parallel to the faces of the primitive rhombohedron. Barytes, which crystallizes in the orthorhombic system, has two sets of cleavages, viz. a single cleavage parallel to the basal pinacoid {001} and also two directions parallel to the faces of the prism {110}. In all of the examples just quoted the cleavage is described as perfect, since cleavage flakes with very smooth and bright surfaces may be readily detached from the crystals. Different substances, however, vary widely in their character of cleavage; in some it can only be described as good or distinct, whilst in others, e.g. quartz and alum, there is little or no tendency to split along certain directions and the surfaces of fracture are very uneven. Cleavage is therefore a character of considerable determinative value, especially for the purpose of distinguishing different minerals.
Another result of the presence in crystals of directions of minimum cohesion are the “percussion figures,” which are produced on a crystal-face when this is struck with a sharp point. A percussion figure consists of linear cracks radiating from the point of impact, which in their number and orientation agree with the symmetry of the face. Thus on a cube face of a crystal of salt the rays of the percussion figure are parallel to the diagonals of the face, whilst on an octahedral face a three-rayed star is developed. By pressing a blunt point into a crystal face a somewhat similar figure, known as a “pressure figure,” is produced. Percussion and pressure figures are readily developed in cleavage sheets of mica (q.v.).
Closely allied to cohesion is the character of “hardness,” which is often defined, and measured by, the resistance which a crystal face offers to scratching. That hardness is a character depending largely on crystalline structure is well illustrated by the two crystalline modifications of carbon: graphite is one of the softest of minerals, whilst diamond is the hardest of all. The hardness of crystals of different substances thus varies widely, and with minerals it is a character of considerable determinative value; for this purpose a scale of hardness is employed (see Mineralogy). Various attempts have been made with the view of obtaining accurate determinations of degrees of hardness, but with varying results; an instrument used for this purpose is called a sclerometer (from σκληρός, hard). It may, however, be readily demonstrated that the degree of hardness on a crystal face varies with the direction, and that a curve expressing these relations possesses the same geometrical symmetry as the face itself. The mineral cyanite is remarkable in having widely different degrees of hardness on different faces of its crystals and in different directions on the same face.
Another result of the differences of cohesion in different directions is that crystals are corroded, or acted upon by chemical solvents, at different rates in different directions. This is strikingly shown when a sphere cut from a crystal, say of calcite
