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Originally appearing in Volume V07, Page 585 of the 1911 Encyclopedia Britannica.
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HEXAGONAL PYRAMIDAL CLASS (Hemimorphic-tetartohedral). No other element is here associated with the hexad axis, which is uniterminal. The pyramids all consist of six faces at one end of the crystal, and prisms are all hexagonal prisms; perpendicular to the hexad axis are the pedions. Lithium potassium sulphate, strontium-antimonyl dextro-tartrate, and lead-antimonyl dextro-tartrate are examples of this type of symmetry. The mineral nepheline is placed in this class because of the absence of symmetry in the etched figures on the prism faces (fig. 92). (g) Regular Grouping of Crystals. Crystals of the same kind when occurring together may some-times be grouped in parallel position and so give rise to special structures, of which the dendritic (from bivipov, a tree) or branch-like aggregations of native copper or of magnetite and the fibrous structures of many minerals furnish examples. Sometimes, owing to changes in the surrounding conditions, the crystal may continue its growth with a different external form or colour, e.g. sceptre-quartz. Regular intergrowths of crystals of totally different substances such as staurolite with cyanite, rutile with haematite, blende with chalcopyrite,calcite with sodium nitrate, are not uncommon. In these cases certain planes and edges of the two crystals are parallel. (See O. Mugge, " Die regelmassigen Verwachsungen von Mineralien verschiedener Art," Neues Jahrbuch fur Mineralogic, 1903, vol. xvi. pp. 335-475.) But by far the most important kind of regular conjunction of crystals is that known as " twinning." Here two crystals or individuals of the same kind have grown together in a certain symmetrical manner, such that one portion of the twin may be brought into the position of the other by reflection across aplane or by rotation about an axis. The plane of reflection is called the twin-plane, and is parallel to one of the faces, or to a possible face, of the crystal: the axis of rotation, called the twin-axis, is parallel to one of the edges or perpendicular to a face of the crystal. In the twinned crystal of gypsum represented in fig. 81 the two portions are symmetrical with respect to a plane parallel to the ortho-pinacoid (too), i.e. a vertical plane perpendicular to the face b. Or we may consider the simple crystal (fig. 82) to be cut in half by this plane and one portion to be rotated through 18o° about the normal to the same plane. Such a crystal (fig. 81) is therefore described as FIG. St.-Twinned FIG. 82.—Simple being twinned on the Crystal of Gypsum. Crystal of Gypsum. plane (too). An octahedron (fig. 83) twinned on an octahedral face (III) has the two portions symmetrical with respect to a plane parallel to this face (the large triangular face in the figure); and either portion may be brought into the position of the other by a rotation through 180° about the triad axis of symmetry which is perpendicular to this face. This kind of twinning is especially frequent in crystals of spinel, and is consequently often referred to as the " spinel twin-law." In these two examples the surface of the union, or composition-plane, of the two portions is a regular surface coinciding with the twin-plane; such twins are called " juxtaposition-twins." In other juxtaposed twins the plane of composition is, however, not necessarily the twin-plane. Another type of twin is the " inter-penetration twin," an example of which is shown in fig. 84. Here one cube may be brought into the position of the other by a rotation of 18o° about a triad axis, or by reflection across the octahedral plane which is perpendicular to this axis; the twin-plane is therefore (iii). Since in many cases twinned crystals may be explained by the rotation of one portion through two right angles, R. J. Hauy introduced the term " hemitrope " (from the Gr. ilµt-, half, and -rptnror, a turn); the word "macle " had been earlier used by Rome d'Isle. There are, however, some rare types of twins which cannot be explained by rotation about an axis, but only by reflection across a plane; these are known as " symmetric twins," a good example of which is furnished by one of the twin-laws of chalcopyrite. Twinned crystals may often be recognized by the presence of re-entrant angles between the faces of the two portions, as may be seen from the above figures. In some twinned crystals (e.g. quartz) there are, however, no re-entrant angles. On the other hand, two crystals accidentally grown together without any symmetrical relation between them will usually show some re-entrant angles, but this must not be taken to indicate the presence of twinning. Twinning may be several times repeated on the same plane or on other similar planes of the crystal, giving rise to triplets, quartets and other complex groupings. When often repeated on the same plane, the twinning is said to be "polysynthetic," and gives rise to a laminated structure in the crystal. Sometimes such a crystal (e.g. of corundum or pyroxene) may be readily broken in this direction, which is thus a " plane of parting," often closely resembling a true cleavage in character. In calcite and some other substances this lamellar twinning may be produced artificially by pressure (see below, Sect. II. (a), Glide-plane). Another curious result of twinning is the production of forms which apparently display a higher degree of symmetry than that actually possessed by the substance. Twins of this kind are known as " mimetic-twins or pseudo-symmetric twins." Two hemihedral or hemimorphic crystals (e.g. of diamond or of hemimorphite) are often united in twinned position to produce a group with apparently the same degree of symmetry as the holosymmetric class of the same system. Or again, a substance crystallizing in, say, the orthorhombic system (e.g. aragonite) may, by twinning, give rise to pseudo-hexagonal forms: and pseudo-cubic forms often result by the complex twinning of crystals (e.g. stannite, phillipsite, &c.) belonging to other systems. Many of the so-called " optical anomalies " of crystals may be explained by this pseudo-symmetric twinning. (h) Irregularities of Growth of Crystals; Character of Faces. Only rarely do actual crystals present the symmetrical appearance shown in the figures given above, in which similar faces are all represented as of equal size. It frequently happens that the crystal is so placed with respect to the liquid in which it grows that there will be a more rapid deposition of material on one part than on another; for instance, if the crystal be attached to some other solid it cannot grow in that direction. Only when a crystal is freely suspended in the mother-liquid and material for growth is supplied at the same rate on all sides does an equably developed form result. Two misshapen or distorted octahedra are represented in figs. 85 and 86; the former is elongated in the direction of one of the edges of the octahedron, and the latter is flattened parallel to one pair of faces. It will be noticed in these figures that the edges in which the faces intersect have the same directions as before, though here there are additional edges not present in fig. 3. The angles (70° 32' or I090 28') between the faces also remain the same; and the faces have the same inclinations to the axes and planes of symmetry as in the equably developed form. Al-though from a geometrical point of view these figures are no Misshapen Octahedra. longer symmetrical with respect to the axes and planes of symmetry, yet crystallographically they are just as symmetrical as the ideally developed form, and, however much their irregularity of development, they still are regular (cubic) octahedra of crystallography. A remarkable case of irregular development is presented by the mineral cuprite, which is often found as well-developed octahedra; but in the variety known as chalcotrichite it occurs as a matted aggregate of delicate hairs, each of which is an individual crystal enormously elongated in the direction of an edge or diagonal of the cube. The symmetry of actual crystals is sometimes so obscured by irregularities of growth that it can only be determined by measurement of the angles. An extreme case, where several of the planes have not been developed at all, is illustrated in fig. 87, which shows the actual shape of a crystal of zircon from Ceylon; the ideally developed form (fig. 88) is placed at the side for corn-parison, and the parallelism of the edges between corresponding faces will be noticed. This crystal is a combination of five simple forms, viz. two tetragonal prisms (a and m,) two tetragonal bipyramids (e and p), and one ditetragonal bipyramid (x, with 16 faces). The actual form, or " habit," of crystals may vary widely in different crystals of the same substance, these differences depending largely on the conditions under which the growth has taken place. The material may have crystallized from a fused Crystal of Zircon (clinographic drawings and plans). mass or from a solution; and in the latter case the solvent may be of different kinds and contain other substances in solution, or the temperature may vary. Calcite (q.v.) affords a good example of a substance crystallizing in widely different habits, but all crystals are referable to the same type of symmetry and may be reduced to the same fundamental form. When crystals are aggregated together, and so interfere with each other's growth, special structures and external shapes often result, which are sometimes characteristic of certain substances, especially amongst minerals. Incipient crystals, the development of which has been arrested owing to unfavourable conditions of growth, are known as crystallites (q.v.). They are met with in imperfectly crystallized substances and in glassy rocks (obsidian and pitchstone), or may be obtained artificially from a solution of sulphur in carbon disulphide rendered viscous by the addition of Canada-balsam. To the various forms H. Vogelsang gave, in 1875, the names "globulites," "margarites" (from i apyapi'njs, a pearl), "longulites," &c. At a more advanced stage of growth these bodies react on polarized light, thus possessing the internal structure of true crystals; they are then called " microlites." These have the form of minute rods, needles or hairs, and are aggregated into feathery and spherulitic forms or skeletal crystals. They are common constituents of microcrystalline igneous rocks, and often occur as inclusions in larger crystals of other substances. Inclusions of foreign matter, accidentally caught up during growth, are frequently present in crystals. Inclusions of other minerals are specially frequent and conspicuous in crystals of quartz, and crystals of calcite may contain as much as 6o % of included sand. Cavities, either with rounded boundaries or with the same shape (" negative crystals ") as the surrounding crystal, are often to be seen; they may be empty or enclose a liquid with a movable bubble of gas. The faces of crystals are rarely perfectly plane and smooth, but are usually striated, studded with small angular elevations, pitted or cavernous, and sometimes curved or twisted. These irregularities, however*, conform with the symmetry of the crystal, and much may be learnt by their study. The parallel grooves or furrows, called " striae," are the result of oscillatory combination between adjacent faces, narrow strips of first one face and then another being alternately developed. Sometimes the striae on crystal-faces are due to repeated lamellar twinning, as in the plagioclase felspars. The directions of the striations are very characteristic features of many crystals: e.g. the faces of the hexagonal prism of quartz are always striated horizontally, whilst in beryl they are striated vertically. Cubes of pyrites (fig. 89) are striated parallel to one edge, the striae on adjacent faces being at right angles, and due to oscillatory combination of the cube and the pentagonal dodecahedron (compare fig. 36) whilst cubes of blende (fig. 90) are striated parallel to one diagonal of each face, i.e. parallel to the tetrahedron faces (compare fig. 31). These striated cubes thus possess different degrees of symmetry and belong to different symmetry-classes. Oscillatory combination of faces gives rise also to curved surfaces. Crystals with twisted surfaces (see DOLOMITE) are, however, built up of smaller crystals arranged in nearly parallel position. Sometimes a face is entirely replaced by small faces of other forms, giving rise to a drusy surface; an example of this is shown by some octahedral crystals of fluorspar (fig. 2) which are built up of minute cubes. The faces of crystals are sometimes partly or completely replaced by smooth bright surfaces inclined at only a few minutes of arc from the true position of the face; such surfaces are called " vicinal faces," and their indices can be expressed only by very high numbers. In apparently perfectly developed crystals of alum the octahedral face, with the simple indices (III), is usually replaced by faces of very low triakis-octahedra, with indices such as (251.251.250); the angles measured on such crystals will therefore deviate slightly from the true octahedral angle. Vicinal faces of this character are formed during the growth of crystals, and have been studied by H. A. Miers (Phil. Trans., 1903, Ser. A. vol. 202). Other faces with high indices, viz. " prerosion faces " and the minute faces forming the sides of etched figures (see below), as well as rounded edges and other surface irregularities, may, however, result from the corrosion of a crystal subsequent to its growth. The pitted and, cavernous faces of artificially grown crystals of sodium chloride and of bismuth are, on the other hand, a result of rapid growth, more material being supplied at the edges and corners of the crystal than at the centres of the faces. (i) Theories of Crystal Structure. The ultimate aim of crystallographic research is to determine the internal structure of crystals from both physical and. chemical data. The problem is essentially twofold: in the first place it is necessary to formulate a theory as to the disposition of the molecules, which conforms with the observed types of symmetry —this is really a mathematical problem; in the second place, it is necessary to determine the orientation of the atoms (or groups of atoms) composing the molecules with regard to the crystal axes—this involves a knowledge of the atomic structure of the molecule. As appendages to the second part of our problem, there have to be considered: (1) the possibility of the existence of the same substance in two or more distinct crystal-line forms—polymorphism, and (2) the relations between the chemical structure of compounds which affect nearly identical or related crystal habits—isomorphism and morphotropy. Here we shall discuss the modern theory of crystal structure; the relations between chemical composition and crystallographical form are. discussed in Part III. of this article; reference should also be made to the article CHEMISTRY: Physical. The earliest theory of crystal structure of any moment is that of Haiiy, in which, as explained above, he conceived a. crystal as composed of elements bounded by the cleavage »aiiy. planes of the crystal, the elements being arranged contiguously and along parallel lines. There is, however, no reason to suppose that matter is continuous throughout a crystalline body; in fact, it has been shown that space does. separate the molecules, and we may therefore replace the contiguous elements of Hauy by particles equidistantly distributed along parallel lines; by this artifice we retain the reticulated or net-like structure, but avoid the continuity of, matter which characterizes Hauy's theory; the permanence of crystal form being due to equilibrium between the inter-molecular (and interatomic) forces. The crystal is thus Conjectured as a " space-lattice," composed of three sets of parallel planes which enclose parallelopipeda, at the corners of which are placed the constituent molecules (or groups of, molecules) of the crystal. The geometrical theory of crystal structure (i.e. the determination of the varieties of crystal symmetry) is thus reduced to the mathematical problem: " in how many ways can space be partitioned ? " M. L. Frankenheim, in 1835, Franken.. determined this number as fifteen, but A. Bravais, Brevrim' als. in 185o, proved the identity of two of Frankenheim's forms, and showed how the remaining fourteen coalesced by pairs, so that really these forms only corresponded to seven distinct systems and fourteen classes of crystal symmetry. These systems, however, only represented holohedral forms, leaving the hemihedral and tetartohedral classes to be explained. Bravais attempted an explanation by attributing differences in the symmetry of the crystal elements, or, what comes to the same thing, he assumed the crystals to exhibit polar differences along any member of the lattice; for instance, assume the particles to be (say) pear-shaped, then the sharp ends point in one direction, the blunt ends in the opposite direction. A different view was adopted by L. Sohncke in 1879, who, by developing certain considerations published by Camille Jordan in 1869 on the possible types of regular repeti- sohncke. tion in space of identical parts, showed that the lattice-structure of Bravais was unnecessary,, it being sufficient that each molecule of an indefinitely extended crystal, represented by its " point " (or centre of gravity), was identically situated with respect to the molecules surrounding it. The problem then resolves itself into the determination of the number of " point-systems " possible; Sohncke derived sixty-five such arrangements, which may also be obtained from the fourteen space-lattices of Bravais, by interpenetrating any one space-lattice with one or more identical lattices, with the condition that the resulting structure should conform with the homogeneity characteristic of crystals. But the sixty-five arrangements derived by Sohncke, of which Bravais' lattices are particular cases, did not complete the solution, for certain of the known types of crystal symmetry still remained unrepresented. These missing forms are characterized as being enantiomorphs consequently, with the introduction of this principle of repetitio over a plane, i.e. mirror images. E. S. Fedorov (189o), A Schoenflies, (1891), and W. Barlow (1894), independently and by different methods, showed how Sohncke's theory of regular point-systems explained the whole thirty-two classes of crystal symmetry, 230. distinct types of crystal structure falling into these classes. By considering the atoms instead of the centres of gravity of the molecules, Sohncke (Zeits. Kryst. Min., 1888, 14, p. 431), has generalized his theory, and propounded the structure of a crystal in the following terms: " A crystal consists of a finite number of interpenetrating regular point-systems, which all possess like and like-directed coincidence movements. Each separate point-system is occupied by similar material particles, but these may be different for the different interpenetrating partial systems which, form the complex system." Or we may quote the words of P. von Groth (British Assoc. Rep., 1904): " A crystal—considered as indefinitely extended—consists of n 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 { too } , diamond and fluorspar cleave in four directions parallel to the octahedral faces 11111, and blende in six directions parallel to the faces of the rhombic dodecahedron { 1 o } . 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 { oo1 } and also two directions parallel to the faces of the prism { 'cal. 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 mini-mum 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 rsXOpbs, hard). It may, however, be readily demonstrated that the degree of hardness on a crystal face varies with the direction, and that a curve ex-pressing 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 crystals may often be readily broken along these directions, f solvents, at different rates in different directions. This is which are thus " planes of parting " or " pseudo-cleavage." strikingly shown when a sphere cut from a crystal, say of calcite 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 resume, with references to the literature, will be found in Report on the Development of the Geometrical Theories of Crystal Structure, 1666–19ot " (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 in-stance, 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, Flilssige Kristalle, Leipzig, r904 ; F. R. Schenck, Kristallinische Fliissigkeiten and fliissige Krystalle, Leipzig, 1905.) 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,
End of Article: HEXAGONAL PYRAMIDAL CLASS (Hemimorphic-tetartohedral)

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