Authorities.—A useful essay is that of “Miss X” (Miss Goodrich Freer) in theProceedings of the Society for Psychical Research, v. The history of crystal-gazing is here traced, and many examples of the author’s own experiments are recorded. A. Lang’sThe Making of Religion, ch. v., contains anthropological examples and a series of experiments. In N. W. Thomas’sCrystal Gazingthe history and anthropology of the subject are investigated, with modern instances. For Egypt, see Lane’sModern Egyptians, and theJournalof Sir Walter Scott, xi. 419-421, withQuarterly Review, No. 117, pp. 196-208. These Egyptian experiments of 1830 were vitiated by their method, the scryer being asked to see and describe a given person, named. He ought not, of course, to be told more than that he is to descry the inquirer’s thoughts, and there ought never to be physical contact, as in holding hands, between the inquirer and the scryer during the experiment. There is a chapter on crystal-gazing inLes Névroses et les idées fixesof Dr Janet (1898). His statements are sometimes demonstrably inaccurate (seeMaking of Religion, Appendix C). A curious passage on the subject, by Ibn Khaldun, an Arabian medievalsavant, is quoted by Mr Thomas from the printed Extracts of MSS. in the Bibliothèque Nationale. There is also a chapter on crystal-gazing in Myers’Human Personality.
Authorities.—A useful essay is that of “Miss X” (Miss Goodrich Freer) in theProceedings of the Society for Psychical Research, v. The history of crystal-gazing is here traced, and many examples of the author’s own experiments are recorded. A. Lang’sThe Making of Religion, ch. v., contains anthropological examples and a series of experiments. In N. W. Thomas’sCrystal Gazingthe history and anthropology of the subject are investigated, with modern instances. For Egypt, see Lane’sModern Egyptians, and theJournalof Sir Walter Scott, xi. 419-421, withQuarterly Review, No. 117, pp. 196-208. These Egyptian experiments of 1830 were vitiated by their method, the scryer being asked to see and describe a given person, named. He ought not, of course, to be told more than that he is to descry the inquirer’s thoughts, and there ought never to be physical contact, as in holding hands, between the inquirer and the scryer during the experiment. There is a chapter on crystal-gazing inLes Névroses et les idées fixesof Dr Janet (1898). His statements are sometimes demonstrably inaccurate (seeMaking of Religion, Appendix C). A curious passage on the subject, by Ibn Khaldun, an Arabian medievalsavant, is quoted by Mr Thomas from the printed Extracts of MSS. in the Bibliothèque Nationale. There is also a chapter on crystal-gazing in Myers’Human Personality.
(A. L.)
1Proceedings of the Society for Psychical Research, v. 486.2“Philosophie der Geistes,” Hegel’sWerke, vii. 179, 406, 408 (Berlin, 1845). Cf. Wallace’s translation (Oxford, 1894).
1Proceedings of the Society for Psychical Research, v. 486.
2“Philosophie der Geistes,” Hegel’sWerke, vii. 179, 406, 408 (Berlin, 1845). Cf. Wallace’s translation (Oxford, 1894).
CRYSTALLITE.In media which, on account of their viscosity, offer considerable resistance to those molecular movements which are necessary for the building and growth of crystals, rudimentary or imperfect forms of crystallization very frequently occur. Such media are the volcanic rocks when they are rapidly cooled, producing various kinds of pitchstone, obsidian, &c. When examined under the microscope these rocks consist largely of a perfectly amorphous or glassy base, through which are scattered great numbers of very minute crystals (microliths), and other bodies, termed crystallites, which seem to be stages in the formation of crystals. Crystallites may also be produced by allowing a solution of sulphur in carbon disulphide mixed with Canada balsam to evaporate slowly, and their development may be watched on a microscopic slide. Small globules appear (globulites), spherical and non-crystalline (so far as can be ascertained). They may coalesce or may arrange themselves into rows like strings of beads—margarites—(Gr.μαργαρίτης, a pearl) or into groups with a somewhat radiate arrangement—globospherites. Occasionally they take elongated shapes—longulites and baculites (Lat.baculus, a staff). The largest may become crystalline, changing suddenly into polyhedral bodies with evident double refraction and the optical properties belonging to crystals. Others become long and thread-like—trichites (Gr.θρίξ, τριχός, hair)—and these are often curved, and a group of them may be implanted on the surface of a small crystal. All these forms are found in vitreous igneous rocks. H. P. J. Vogelsang, who was the first to direct much attention to them, believes that the globulites are preliminary stages in the formation of crystals.
Microliths, as distinguished from crystallites, have crystalline properties, and evidently belong to definite minerals or salts. When sufficiently large they are often recognizable, but usuallythey are so small, so opaque, or so densely crowded together that this is impossible. In igneous rocks they are usually felspar, augite, enstatite, and iron oxides, and are found in abundance only where there is much uncrystallized glassy base; in contact-altered sediments, slags, &c., microlithic forms of garnet, spinel, sillimanite, cordierite, various lime silicates, and many other substances have been observed. Their form varies greatly,e.g.thin fibres (sillimanite, augite), short prisms or rods (felspar, enstatite, cordierite), or equidimensional grains (augite, spinel, magnetite). Occasionally they are perfectly shaped though minute crystals; more frequently they appear rounded (magnetite, &c.), or have brush-like terminations (augite, felspar, &c.). The larger microliths may contain enclosures of glass, and it is very common to find that the prisms have hollow, funnel-shaped ends, which are filled with vitreous material. These microliths, under the influence of crystalline forces, may rank themselves side by side to make up skeleton crystals and networks, or feathery and arborescent forms, which obey more or less closely the laws of crystallization of the substance to which they belong. They bear a very close resemblance to the arborescent frost flowers seen on window panes in winter, and to the stellate snow crystals. In magnetite the growths follow three axes at right angles to one another; in augite this is nearly, though not exactly, the case; in hornblende an angle of 57° may frequently be observed, corresponding to the prism angle of the fully-developed crystal. The interstices of the network may be partly filled up by a later growth. In other cases the crystalline arrangement of the microliths is less perfect, and branching, arborescent or feathery groupings are produced (e.g.felspar, augite, hornblende). Spherulites may be regarded as radiate aggregates of such microliths (mostly felspar mixed with quartz or tridymite). If larger porphyritic crystals occur in the rock, the microliths of the vitreous base frequently grow outwards from their faces; in some cases a definite parallelism exists between the two, but more frequently the early crystal has served merely as a centre, or nucleus, from which the microliths and spherulites have spread in all directions.
(J. S. F.)
CRYSTALLIZATION,the art of obtaining a substance in the form of crystals; it is an important process in chemistry since it permits the purification of a substance, or the separation of the constituents of a mixture. Generally a substance is more soluble in a solvent at a high temperature than at a low, and consequently, if a boiling concentrated solution be allowed to cool, the substance will separate in virtue of the diminished solubility, and the slower the cooling the larger and more perfect will be the crystals formed. If, as sometimes appears, such a solution refuses to crystallize, the expedient of inoculating the solution with a minute crystal of the same substance, or with a similar substance, may be adopted; shaking the solution, or the addition of a drop of another solvent, may also occasion the desired result. “Fractional crystallization” consists in repeatedly crystallizing a salt so as to separate the substances of different solubilities. Examples are especially presented in the study of the rare-earths. Other conditions under which crystals are formed are given in the articleCrystallography.
CRYSTALLOGRAPHY(from the Gr.κρύσταλλος, ice, andγράφειν, to write), the science of the forms, properties and structure of crystals. Homogeneous solid matter, the physical and chemical properties of which are the same about every point, may be either amorphous or crystalline. In amorphous matter all the properties are the same in every direction in the mass; but in crystalline matter certain of the physical properties vary with the direction. The essential properties of crystalline matter are of two kinds, viz. the general properties, such as density, specific heat, melting-point and chemical composition, which do not vary with the direction; and the directional properties, such as cohesion and elasticity, various optical, thermal and electrical properties, as well as external form. By reason of the homogeneity of crystalline matter the directional properties are the same in all parallel directions in the mass, and there may be a certain symmetrical repetition of the directions along which the properties are the same.
When the crystallization of matter takes place under conditions free from outside influences the peculiarities of internal structure are expressed in the external form of the mass, and there results a solid body bounded by plane surfaces intersecting in straight edges, the directions of which bear an intimate relation to the internal structure. Such a polyhedron (πολύς, many,ἕδρα, base or face) is known as a crystal. An example of this is sugar-candy, of which a single isolated crystal may have grown freely in a solution of sugar. Matter presenting well-defined and regular crystal forms, either as a single crystal or as a group of individual crystals, is said to be crystallized. If, on the other hand, crystallization has taken place about several centres in a confined space, the development of plane surfaces may be prevented, and a crystalline aggregate of differently orientated crystal-individuals results. Examples of this are afforded by loaf sugar and statuary marble.
After a brief historical sketch, the more salient principles of the subject will be discussed under the following sections:—
I.Crystalline Form.(a) Symmetry of Crystals.(b) Simple Forms and Combinations of Forms.(c) Law of Rational Indices.(d) Zones.(e) Projection and Drawing of Crystals.(f) Crystal Systems and Classes.1. Cubic System.2. Tetragonal System.3. Orthorhombic System.4. Monoclinic System.5. Anorthic System.6. Hexagonal System(g) Regular Grouping of Crystals (Twinning, &c.).(h) Irregularities of Growth of Crystals: Characters of Faces.(i) Theories of Crystal Structure.II.Physical Properties of Crystals.(a) Elasticity and Cohesion (Cleavage, Etching, &c.).(b) Optical Properties (Interference figures, Pleochroism, &c.).(c) Thermal Properties.(d) Magnetic and Electrical Properties.III.Relations between Crystalline Form and Chemical Composition.
I.Crystalline Form.
(a) Symmetry of Crystals.
(b) Simple Forms and Combinations of Forms.
(c) Law of Rational Indices.
(d) Zones.
(e) Projection and Drawing of Crystals.
(f) Crystal Systems and Classes.
1. Cubic System.
2. Tetragonal System.
3. Orthorhombic System.
4. Monoclinic System.
5. Anorthic System.
6. Hexagonal System
(g) Regular Grouping of Crystals (Twinning, &c.).
(h) Irregularities of Growth of Crystals: Characters of Faces.
(i) Theories of Crystal Structure.
II.Physical Properties of Crystals.
(a) Elasticity and Cohesion (Cleavage, Etching, &c.).
(b) Optical Properties (Interference figures, Pleochroism, &c.).
(c) Thermal Properties.
(d) Magnetic and Electrical Properties.
III.Relations between Crystalline Form and Chemical Composition.
Most chemical elements and compounds are capable of assuming the crystalline condition. Crystallization may take place when solid matter separates from solution (e.g.sugar, salt, alum), from a fused mass (e.g.sulphur, bismuth, felspar), or from a vapour (e.g.iodine, camphor, haematite; in the last case by the interaction of ferric chloride and steam). Crystalline growth may also take place in solid amorphous matter, for example, in the devitrification of glass, and the slow change in metals when subjected to alternating stresses. Beautiful crystals of many substances may be obtained in the laboratory by one or other of these methods, but the most perfectly developed and largest crystals are those of mineral substances found in nature, where crystallization has continued during long periods of time. For this reason the physical science of crystallography has developed side by side with that of mineralogy. Really, however, there is just the same connexion between crystallography and chemistry as between crystallography and mineralogy, but only in recent years has the importance of determining the crystallographic properties of artificially prepared compounds been recognized.
History.—The word “crystal” is from the Gr.κρύσταλλος, meaning clear ice (Lat.crystallum), a name which was also applied to the clear transparent quartz (“rock-crystal”) from the Alps, under the belief that it had been formed from water by intense cold. It was not until about the 17th century that the word was extended to other bodies, either those found in nature or obtained by the evaporation of a saline solution, which resembled rock-crystal in being bounded by plane surfaces, and often also in their clearness and transparency.
The first important step in the study of crystals was made by Nicolaus Steno, the famous Danish physician, afterwards bishop of Titiopolis, who in his treatiseDe solido intra solidum naturaliter contento(Florence, 1669; English translation, 1671) gave theresults of his observations on crystals of quartz. He found that although the faces of different crystals vary considerably in shape and relative size, yet the angles between similar pairs of faces are always the same. 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 nucleus, this nucleus having the form of a regular six-sided prism terminated at each end by a six-sided pyramid. The thickness of the layers, though the same over each face, was not necessarily the same on different faces, but depended on the position of the faces with respect to the surrounding liquid; hence the faces of the crystal, though variable in shape and size, remained parallel to those of the nucleus, and the angles between them constant. Robert Hooke in hisMicrographia(London, 1665) had previously noticed the regularity of the minute quartz crystals found lining the cavities of flints, and had suggested that they were built up of spheroids. About the same time the double refraction and perfect rhomboidal cleavage of crystals of calcite or Iceland-spar were studied by Erasmus Bartholinus (Experimenta crystalli Islandici disdiaclastici, Copenhagen, 1669) and Christiaan Huygens (Traité de la lumière, Leiden, 1690); the latter supposed, as did Hooke, that the crystals were built up of spheroids. In 1695 Anton van Leeuwenhoek observed under the microscope that different forms of crystals grow from the solutions of different salts. Andreas Libavius had indeed much earlier, in 1597, pointed out that the salts present in mineral waters could be ascertained by an examination of the shapes of the crystals left on evaporation of the water; and Domenico Guglielmini (Riflessioni filosofiche dedotte dalle figure de’ sali, Padova, 1706) asserted that the crystals of each salt had a shape of their own with the plane angles of the faces always the same.
The earliest treatise on crystallography is theProdromus Crystallographiaeof M. A. Cappeller, published at Lucerne in 1723. Crystals were mentioned in works on mineralogy and chemistry; for instance, C. Linnaeus in hisSystema Naturae(1735) described some forty common forms of crystals amongst minerals. It was not, however, until the end of the 18th century that any real advances were made, and the French crystallographers Romé de l’Isle and the abbé Haüy are rightly considered as the founders of the science. J. B. L. de Romé de l’Isle (Essai de cristallographie, Paris, 1772;Cristallographie, ou description des formes propres à tous les corps du règne minéral, Paris, 1783) made the important discovery that the various shapes of crystals of the same natural or artificial substance are all intimately related to each other; and further, by measuring the angles between the faces of crystals with the goniometer (q.v.), he established the fundamental principle that these angles are always the same for the same kind of substance and are characteristic of it. Replacing by single planes or groups of planes all the similar edges or solid angles of a figure called the “primitive form” he derived other related forms. Six kinds of primitive forms were distinguished, namely, the cube, the regular octahedron, the regular tetrahedron, a rhombohedron, an octahedron with a rhombic base, and a double six-sided pyramid. Only in the last three can there be any variation in the angles: for example, the primitive octahedron of alum, nitre and sugar were determined by Romé de l’Isle to have angles of 110°, 120° and 100° respectively. René Just Haüy in hisEssai d’une théorie sur la structure des crystaux(Paris, 1784; see also his Treatises on Mineralogy and Crystallography, 1801, 1822) supported and extended these views, but took for his primitive forms the figures obtained by splitting crystals in their directions of easy fracture of “cleavage,” which are aways the same in the same kind of substance. Thus he found that all crystals of calcite, whatever their external form (see, for example, figs. 1-6 in the articleCalcite), could be reduced by cleavage to a rhombohedron with interfacial angles of 75°. Further, by stacking together a number of small rhombohedra of uniform size he was able, as had been previously done by J. G. Gahn in 1773, to reconstruct the various forms of calcite crystals. Fig. 1 shows a scalenohedron (σκαληνός, uneven) built up in this manner of rhombohedra; and fig. 2 a regular octahedron built up of cubic elements, such as are given by the cleavage of galena and rock-salt.
The external surfaces of such a structure, with their step-like arrangement, correspond to the plane faces of the crystal, and the bricks may be considered so small as not to be separately visible. By making the steps one, two or three bricks in width and one, two or three bricks in height the various secondary faces on the crystal are related to the primitive form or “cleavage nucleus” by a law of whole numbers, and the angles between them can be arrived at by mathematical calculation. By measuring with the goniometer the inclinations of the secondary faces to those of the primitive form Haüy found that the secondary forms are always related to the primitive form on crystals of numerous substances in the manner indicated, and that the width and the height of a step are always in a simple ratio, rarely exceeding that of 1 : 6. This laid the foundation of the important “law of rational indices” of the faces of crystals.
The German crystallographer C. S. Weiss (De indagando formarum crystallinarum charactere geometrico principali dissertatio, Leipzig, 1809;Übersichtliche Darstellung der verschiedenen natürlichen Abtheilungen der Krystallisations-Systeme, Denkschrift der Berliner Akad. der Wissensch., 1814-1815) attacked the problem of crystalline form from a purely geometrical point of view, without reference to primitive forms or any theory of structure. The faces of crystals were considered by their intercepts on co-ordinate axes, which were drawn joining the opposite corners of certain forms; and in this way the various primitive forms of Haüy were grouped into four classes, corresponding to the four systems described below under the names cubic, tetragonal, hexagonal and orthorhombic. The same result was arrived at independently by F. Mohs, who further, in 1822, asserted the existence of two additional systems with oblique axes. These two systems (the monoclinic and anorthic) were, however, considered by Weiss to be only hemihedral or tetartohedral modifications of the orthorhombic system, and they were not definitely established until 1835, when the optical characters of the crystals were found to be distinct. A system of notation to express the relation of each face of a crystal to the co-ordinate axes of reference was devised by Weiss, and other notations were proposed by F. Mohs, A. Lévy (1825), C. F. Naumann (1826), and W. H. Miller (Treatise on Crystallography, Cambridge, 1839). For simplicity and utility in calculation the Millerian notation, which was first suggested by W. Whewell in 1825, surpasses all others and is now generally adopted, though those of Lévy and Naumann are still in use.
Although the peculiar optical properties of Iceland-spar had been much studied ever since 1669, it was not until much later that any connexion was traced between the optical characters of crystals and their external form. In 1818 Sir David Brewster found that crystals could be divided optically into three classes, viz. isotropic, uniaxial and biaxial, and that these classes corresponded with Weiss’s four systems (crystals belonging to the cubic system being isotropic, those of the tetragonal and hexagonal being uniaxial, and the orthorhombic being biaxial). Optically biaxial crystals were afterwards shown by J. F. W. Herschel and F. E. Neumann in 1822 and 1835 to be of three kinds, corresponding with the orthorhombic, monoclinic andanorthic systems. It was, however, noticed by Brewster himself that there are many apparent exceptions, and the “optical anomalies” of crystals have been the subject of much study. The intimate relations existing between various other physical properties of crystals and their external form have subsequently been gradually traced.
The symmetry of crystals, though recognized by Romé de l’Isle and Haüy, in that they replaced all similar edges and corners of their primitive forms by similar secondary planes, was not made use of in defining the six systems of crystallization, which depended solely on the lengths and inclinations of the axes of reference. It was, however, necessary to recognize that in each system there are certain forms which are only partially symmetrical, and these were described as hemihedral and tetartohedral forms (i.e.ἡμι-, half-faced, andτέταρτος, quarter-faced forms).
As a consequence of Haüy’s law of rational intercepts, or, as it is more often called, the law of rational indices, it was proved by J. F. C. Hessel in 1830 that thirty-two types of symmetry are possible in crystals. Hessel’s work remained overlooked for sixty years, but the same important result was independently arrived at by the same method by A. Gadolin in 1867. At the present day, crystals are considered as belonging to one or other of thirty-two classes, corresponding with these thirty-two types of symmetry, and are grouped in six systems. More recently, theories of crystal structure have attracted attention, and have been studied as purely geometrical problems of the homogeneous partitioning of space.
The historical development of the subject is treated more fully in the articleCrystallographyin the 9th edition of this work. Reference may also be made to C. M. Marx,Geschichte der Crystallkunde(Karlsruhe and Baden, 1825); W. Whewell,History of the Inductive Sciences, vol. iii. (3rd ed., London, 1857); F. von Kobell,Geschichte der Mineralogie von 1650-1860(München, 1864); L. Fletcher,An Introduction to the Study of Minerals(British Museum Guide-Book); L. Fletcher,Recent Progress in Mineralogy and Crystallography[1832-1894] (Brit. Assoc. Rep., 1894).
The historical development of the subject is treated more fully in the articleCrystallographyin the 9th edition of this work. Reference may also be made to C. M. Marx,Geschichte der Crystallkunde(Karlsruhe and Baden, 1825); W. Whewell,History of the Inductive Sciences, vol. iii. (3rd ed., London, 1857); F. von Kobell,Geschichte der Mineralogie von 1650-1860(München, 1864); L. Fletcher,An Introduction to the Study of Minerals(British Museum Guide-Book); L. Fletcher,Recent Progress in Mineralogy and Crystallography[1832-1894] (Brit. Assoc. Rep., 1894).
I. CRYSTALLINE FORM
The fundamental laws governing the form of crystals are:—
1. Law of the Constancy of Angle.
2. Law of Symmetry.
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 line or axis, and a plane; these “elements of symmetry” are spoken of as a centre of symmetry, an axis of symmetry, and a plane of symmetry respectively.
Centre of Symmetry.—Crystals which are centro-symmetrical have their faces arranged in parallel pairs; and the two parallel faces, situated on opposite sides of the centre (Oin fig. 3) are alike in surface characters, such as lustre, striations, and figures of corrosion. An octahedron (fig. 3) is bounded by four pairs of parallel faces. Crystals belonging to many of the hemihedral and tetartohedral classes of the six systems of crystallization are devoid of a centre of symmetry.
Axes of Symmetry.—Consider the vertical axis joining the opposite corners a3and ā3of an octahedron (fig. 3) and passing through its centreO: by rotating the crystal about this axis through a right angle (90°) it reaches a position such that the orientation of its faces is the same as before the rotation; the face ā1ā2ā3, for example, coming into the position of a1ā2a3. During a complete rotation of 360° (= 90° × 4), the crystal occupies four such interchangeable positions. Such an axis of symmetry is known as a tetrad axis of symmetry. Other tetrad axes of the octahedron are a2ā2and a1a1.
An axis of symmetry of another kind is that which passing through the centreOis normal to a face of the octahedron. By rotating the crystal about such an axisOp(fig. 3) through an angle of 120° those faces which are not perpendicular to the axis occupy interchangeable positions; for example, the face a1a3a2comes into the position of ā2a1ā3, and ā2a1ā3to a3ā2ā1. During a complete rotation of 360° (= 120° × 3) the crystal occupies similar positions three times. This is a triad axis of symmetry; and there being four pairs of parallel faces on an octahedron, there are four triad axes (only one of which is drawn in the figure).
An axis passing through the centreOand the middle points d of two opposite edges of the octahedron (fig. 4),i.e.parallel to the edges of the octahedron, is a dyad axis of symmetry. About this axis there may be rotation of 180°, and only twice in a complete revolution of 360° (= 180° × 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 diagonal) axes (dd).
Although not represented in the cubic system, there is still another kind of axis of symmetry possible in crystals. This is the hexad axis or hexagonal axis, for which the angle of rotation is 60°, or one-sixth of 360°. There can be only one hexad axis of symmetry in any crystal (see figs. 77-80).
Planes of Symmetry.—A regular octahedron can be divided into two equal and similar halves by a plane passing through the corners a1a3ā1ā3and the centreO(fig. 3). One-half is the mirror reflection of the other in this plane, which is called a plane of symmetry. 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 a1a2ā1ā2and a2a3ā2ā3. 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 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 a3and ā3, the middle pointsdof the edges a1ā2and ā1a2, and the centreO(fig. 4). This is called a dodecahedral plane of symmetry, being parallel to the face of the rhombic dodecahedron which truncates the edge a1a2(compare fig. 14, showing a combination of the octahedron and rhombic dodecahedron). Another similar plane of symmetry is that passing through the corners a3ā3and the middle points of the edges a1a2and ā1ā2, and altogether there are six dodecahedral planes of symmetry, two through each of the corners a1, a2, a3of the octahedron.
A regular octahedron and a cube are thus each symmetrical with respect to the following elements of symmetry: a centre of symmetry, thirteen axes of symmetry (of three kinds), and nine planes of symmetry (of two kinds). This degree of symmetry, which is the type corresponding to one of the classes of the cubic system, is the highest possible in crystals. As will be pointed out below, it is possible, however, for both the octahedron and the cube to be associated with fewer elements of symmetry than those just enumerated.
(b)Simple Forms and Combinations of Forms.
A single face a1a2a3(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 a3ā3the four upper faces are obtained; and by rotation of these about one or other of the horizontal tetrad axes the eight faces are derived. Or again, the same repetition of the faces may be arrived at by reflection across the three cubic planes of symmetry. (By reflection across the six dodecahedral planes of symmetry a tetrahedron only would result, but if this is associated with a centre of symmetry we obtain the octahedron.) Such a set of similar faces, obtained by symmetrical repetition, constitutes a “simple form.” An octahedron thus consists of eight similar faces, and a cube is bounded by six faces all of which have the same surface characters, and parallel to each of which all the properties of the crystal are identical.
Examples of simple forms amongst crystallized substances are octahedra of alum and spinel and cubes of salt and fluorspar. More usually, however, two or more forms are present on a crystal, and we then have a combination of forms, or simply a “combination.” Figs. 6, 7 and 8 represent combinations of the octahedron and the cube; in the first the faces of the cube predominate, and in the third those of the octahedron; fig. 7 with the two forms equally developed is called a cubo-octahedron. Each of these combined forms has all the elements of symmetry proper to the simple forms.
The simple forms, though referable to the same type of symmetry and axes of reference, are quite independent, and cannot be derived one from the other by symmetrical repetition, but, after the manner of Romé de l’Isle, they may be derived by replacing edges or corners by a face equally inclined to the faces forming the edges or corners; this is known as “truncation” (Lat.truncare, to cut off). Thus in fig. 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 lie the axial planes. A fourth face on the crystal intersecting these three axes in the pointsA,B,Cis taken as the parametral plane, and the lengthsOA:OB:OCare the parameters of the crystal. Any other face on the crystal may be referred to these axes and parameters by the ratio of the intercepts
Thus for a face parallel to the plane A Be the intercepts are in the ratioOA:OB:Oe, or
and for a planefgCthey areOf:Og:OCor
Now the important relation existing between the faces of a crystal is that the denominatorsh, kandlare always rational whole numbers, rarely exceeding 6, and usually 0, 1, 2 or 3. Written in the form (hkl),hreferring to the axisOX,ktoOY, andltoOZ, they are spoken of as the indices (Millerian indices) of the face. Thus of a face parallel to the planeABCthe indices are (111), ofABethey are (112), and offgC(231). 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 indices will be simpler and all faces of the same simple form will have a similar set of indices. The angles between the axes and the ratio of the lengths of the parametersOA : OB : OC(usually given asa : 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 (111) 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 (111), (111), (111), (111), (111), (111), (111), (111). The symbol {111} indicates all the faces belonging to this simple form. The indices of the six faces of the cube are (100), (010), (001), (100), (010), (001); here each face is parallel to two axes,i.e.intercepts them at infinity, so that the corresponding indices are zero.
(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 centreOof 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-facesa, m, &c. constitute a zone, denotedby [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], whereu= kl1− lk1,v= lh1− hl1, andw= 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.
The kind of projection most extensively used is the “stereographic projection.” The crystal is considered to be placed inside a sphere from the centre of which normals are drawn to all the faces of the crystal. The points at which these normals intersect the surface of the sphere are called the poles of the faces, and by these poles the positions of the faces are fixed. The poles of all faces in the same zone on the crystal will lie on a great circle of the sphere, which are therefore called zone-circles. The calculation of the angles between the normals of faces and between zone-circles is then performed by the ordinary methods of spherical trigonometry. The stereographic projection, however, represents the poles and zone-circles on a plane surface and not on a spherical surface. This is achieved by drawing lines joining all the poles of the faces with the north or south pole of the sphere and finding their points of intersection with the plane of the equatorial great circle, or primitive circle, of the sphere, the projection being represented on this plane. In fig. 10 is shown the stereographic projection, or stereogram, of a cubic crystal; a1, a2, &c. are the poles of the faces of the cube. o1, o2, &c. those of the octahedron, and d1, 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. 11, 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 [a1o1d5], [a1d1a5], [a5o1d2], &c. are zones. A stereographic projection of a rhombohedral crystal is given in fig. 72.
Another kind of projection in common use is the “gnomonic projection” (fig. 12). Here the plane of projection is tangent to the sphere, and normals to all the faces are drawn from the centre of the sphere to intersect the plane of projection. In this 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 (111), 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 geometry (e.g.the plans in the lower part of figs. 87 and 88) have sometimes the advantage of showing the symmetry of a crystal, but they give no idea of solidity. For instance, a cube would be represented merely by a square, and an octahedron by a square with lines joining the opposite corners. True perspective drawings are never used in the representation of crystals, since for showing the zonal relations it is important to preserve the parallelism of the edges. If, however, the eye, or point of vision, is regarded as being at an infinite distance from the object all the rays will be parallel, and edges which are parallel on the crystal will be represented by parallel lines in the drawing. The plane of the drawing, in which the parallel rays joining the corners of the crystals and the eye intersect, may be either perpendicular or oblique to the rays; in the former case we have an “orthographic” (ὀρθός, straight;γράφειν, to draw) drawing, and in the latter a “clinographic” (κλίνειν, to incline) drawing. Clinographic drawings are most frequently used for representing crystals. In representing, for example, a cubic crystal (fig. 11) a cube face a5is 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 a2comes into view,and the eye is then raised so that a third cube face a1may 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.