Crystallography
From LoveToKnow 1911
CRYSTALLOGRAPHY (from the Gr. KpuaraAAos, ice, and 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 (7roXi s, many, g 8pa, 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.
I. 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.
(z) 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.
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History
The word " crystal " is from the Gr. KpuvTaXAos, 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 treatise De solido intra solidum naturaliter contento (Florence, 1669; English translation, 1671) gave the results 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 his Micrographia (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 (Traite de la lumiere, 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 the Prodromus Crystallographiae of M. A. Cappeller, published at Lucerne in 1723. Crystals were mentioned in works on mineralogy and chemistry; for instance, C. Linnaeus in his Systema 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 Rome de l'Isle and the abbe Ha-ay are rightly considered as the founders of the science. B. L. de Rome de l'Isle (Essai de cristallographie, Paris, 1772; Cristallographie, ou description des formes propres ¢ tous les corps du regne mineral, 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, 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 Rome de l'Isle to have angles of 110°, and loo° respectively. Rene Just Haiiy in his Essai d'une theorie 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 article Calcite), 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 1 773, to reconstruct the various forms of calcite crystals. Fig. 1 shows a scalenohedron ,(oKaXnvos, 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 FIG. i. - Scalenohedron built up of Rhombohedra.
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 Haiiy 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 I: 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; Obersichtliche Darstellung der verschiedenen natiirlichen 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 'fatly 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. Levy (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 Levy 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 and FIG. 2. - Octahedron built up of Cubes.
anorthic 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 Rome de l'Isle and Haiiy, 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. nµc-, half-faced, and TETapTOS, quarter-faced forms).
As a consequence of Haiiy'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 article Crystallography in 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 (Munchen, 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: I. 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 (0 in 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 a 3 and a 3 of an octahedron (fig. 3) and passing through its centre 0: 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 a l d,d 3, for example, coming into the position of a1a2a3. During a complete rotation of 360 (= 90°X 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 a 2 a 2 and alai.
An axis of symmetry of another kind is that which passing through the centre 0 is normal to a face of the octahedron. By rotating the crystal about such an axis Op (fig. 3) through an angle of 120° those faces which are not perpendicular to the axis occupy interchangeable positions; for example, the face a 1 a 3 a 2 comes into the position of a 2 a 1 a 3, and a 2 a 1 a 3 to a3a2a1. During a complete rotation of 360 (=120° X3) 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 centre 0 and 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 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 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 a l a 3 a 1 a 3 and the centre 0 (fig. 3). One-half is the mirror reflection of the other in this plane, which p 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 ala2ala2 and a2a3a2a3. These three similar planes of P 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 a 3 and a 3, the middle points d of the edges a 1 a 2 and aia2, and the centre 0 (fig. 4). This is called a dodecahedral plane of symmetry, being parallel to the face of the rhombic dodecahedron which truncates the edge ala2 (compare fig. 14, showing a combination of the octahedron and rhombic dodecahedron). Another similar plane of symmetry is that passing through the corners a 3 a 3 and the middle points of the edges a l a 2 and aid-2, and altogether there are six dodecahedral planes of symmetry, two through each of the corners a l, a2, a 3 of the octahedron.
a 3 R3 FIG. 3. FIG. 4. Axes and Planes of Symmetry of an Octahedron.
FIG. 5. - Axes of Symmetry of a Cube.
p 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 a l a 2 a 3 (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 a 3 a 3 the 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 FIG. 6. - Cube in combination with Octahedron.
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 Rome de l'Isle, they may be derived by replacing edges or corners by a face equally combination with Cube. 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 points A, B, C is taken as the parametral plane, and the lengths OA: OB :OC are 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 OA, OB. OC h k l Thus for a face parallel to the plane ABe the intercepts are in the ratio or OA. OB. OC 'I I 2 and for a plane fgC they are Of: Og: OC or OA, OB, 0 C 2 3 i Now the important relation existing between the faces of a crystal is that the denominators h, k and l are always rational whole numbers, rarely exceeding 6, and usually o, I, 2 or 3. Written in the form (hkl), h referring to the axis OX, k to OY, and 1 to OZ, they are spoken of as the indices (Millerian indices) of the face. Thus of a face parallel to the plane ABC the indices are (III), of ABe they are (112), and of fgC (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 Z indices. The angles between FIG. 9. - Crystallographic axe's 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), (III), MI), (III), (III), (III), (iii). The symbol {III} indicates all the faces belonging to this simple form. The indices of the six faces of the cube are (too), (oio), (ooi), (Too), (oio), (ooi); 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 centre 0 of 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-faces a, m, &c. constitute a zone, denoted FIG. 7. - Cubo-octahedron." FIG. 8. - Octahedron in Y by [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 (h i k,l,) is given by the zone-indices [uvw], where u=kl 1 - lk 1, v=lh 1 - hl 1 , and 'w=' hk 1 - kh 1 , these being obtained from the face-indices by cross multiplication as follows: - h k 1 h k l xx x h, k, 1, h, k111. Any other face (h 2 k 2 l 2 ) lying in this zone must satisfy the equation h2u kiv +1 2 w= o.
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, (31 i) in fig. 12 lies at the intersections of the three zones [210, 'oil, [201, and [211, ioo], 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 FIG. io. - Stereographic Projection of a Cubic Crystal.
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. z o is shown the stereographic projection, or stereogram, of a cubic crystal; a', a 2 , &c. are the poles of the faces of the cube, o l, 0 2 , & c. those of the octahedron, and d', d 2 , & 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. 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 ], [ ald ' a5 ], [a 5 o 1 d 2 ], &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 FIG. I I. - Clinodrawn from the centre of the sphere to graphic Drawing of a intersect the plane of projection. 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 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 " (ip86s, straight; Xpit4 iv, to draw) drawing, and in the latter a " clinographic " (aLPEuv, to incline) of ? J 02J ff E 532 ? 3 ff5' 4?f ?3J ?P 1 0 430340 1 3J J FIG. 12. - Gnomonic Projection of a Cubic Crystal.
drawing. Clinographic drawings are most frequently used for representing crystals. In representing, for example, a cubic crystal (fig. ii) a cube face a' 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 a 2 comes into view, 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.
In crystals of this system the angle between any two faces P and Q with the indices (hkl) and (pqr) is given by the equation Cos r PQ= hp-1-44-1 Holosymmetric Class (Holohedral (6Xos, whole); Hexakis-octahedral).
Crystals of this class possess the full number of elements of symmetry already mentioned above for the octahedron and the cube, viz. three cubic planes of symmetry, six dodecahedral planes, three tetrad axes of symmetry, four triad axes, six dyad axes, and a centre of symmetry.
FIG. 14. - Combination of Rhombic Dodecahedron and Octahedron.
There are seven kinds of simple forms, viz.: Cube (fig. 5). This is bounded by six square faces parallel to the cubic planes of symmetry; it is known also as the hexahedron. The angles between the faces are 90°, and the indices of the form are {loo}. Salt, fluorspar and galena crystallize in simple cubes.
'FIG. 15
Triakis-octahedron. FIG. 16. - Combination of Triakisoctahedron and Cube.
Octahedron (fig. 3). Bounded by eight equilateral triangular faces perpendicular to the triad axes of symmetry. The angles between the faces are 70° 32' and 109° 28', and the indices are {iii }. Spinel, magnetite and gold crystallize in simple octahedra. Combinations of the cube and octahedron are shown in figs. 6-8.
Rhombic dodecahedron (fig. 13). Bounded by twelve rhombshaped faces parallel to the six dodecahedral planes of symmetry. The angles between the normals to adjacent faces are 60°, and between other pairs of faces 90°; the indices are !Ho). Garnet frequently crystallizes in this form. Fig. 14 shows the rhombic dodecahedron in combination with the octahedron.
In these three simple forms of the cubic system (which are shown in combination in fig. I I) the angles between the faces and the indices FIG. 18. - Combination of Icositetrahedron and Cube.
are fixed and are the same in all crystals; in the four remaining simple forms they are variable.
Triakis-octahedron (three-faced octahedron) (fig. 15). This solid is bounded by twenty-four isosceles triangles, and may be considered as an octahedron with a low triangular pyramid on each of its faces. As the inclinations of the faces may vary there is a series of these forms with the indices 12211, 13311, {3321, &c. or in general {hhk}. FIG. 20. - Combination of Icositetrahedron 121 11 and Rhombic Dodecahedron.
Icositetrahedron (fig. 17). Bounded by twenty-four trapezoidal faces, and hence sometimes called a " trapezohedron." The indices are 12111, 1311}, 13221, &c., or in general {hkk} . Analcite, leucite and garnet often crystallize in the simple form {211}. Combinations are shown in figs. 18-20. The plane ABe in fig. 9 is one face (112) of an icositetrahedron; the indices of the remaining faces in this octant being (211) and (121).
FIG. 21. - Tetrakis-hexahedron. FIG. 22. - Tetrakis-hexahedron.
Tetrakis-hexahedron (four-faced cube) (figs. 21 and 22). Like the triakis-octahedron this solid is also bounded by twenty-four isosceles triangles, but here grouped in fours over the cubic faces. The two figures show how, with different inclinations of the faces, the form may vary, approximating in fig. 21 to the cube and in fig. 22 to the rhombic dodecahedron. The angles over the edges lettered A are different from the angles over the edges lettered C. Each face is parallel to one of the crystallographic axes and intercepts the two others in different lengths; the indices are therefore {210}, {310}, {320}, &c., in general {hko} . Fluorspar some times crystallizes in the simple form 13101; more usually, however, in combination with the cube (fig. 23).
Hexakis-octahedron (fig. 24). Here each face of the octahedron is replaced by six scalene triangles,. so that altogether there are FIG. 13. - Rhombic Dodecahedron.
FIG. 17. - Icositetrahedron.
FIG. 19. - Combination of Icositetrahedron and Octahedron.
J(h2+k2+12) (p2+g2+r2).
The angles between faces with the same indices are thus the same in all substances which crystallize in the cubic system: in other systems the angles vary with the substance and are characteristic of it.
FIG. 23. - Combination of Tetrakis-hexahedron and Cube.
forty-eight faces. This is the greatest number of faces possible for any simple form in crystals. The faces are all oblique to the planes and axes of symmetry, and they intercept the three crystallographic axes in different lengths, hence the indices are all unequal, being in general {hkl}, or in particular cases 13211, 14211, {4321, &c. Such a form is known as the " general form " of the class. The interfacial angles over the three edges of each triangle are all different. These forms usually exist only in combination with other cubic forms (for example, fig. 25), but {421}has been observed as a simple form on fluorspar.
Several examples of substances which crystallize in this class have been mentioned above under the different forms; many others might be cited - for instance, the metals iron, copper, silver, gold, platinum, lead, mercury, and the non-metallic elements silicon and phosphorus.
Tetrahedral Class (Tetrahedral-hemihedral; Hexakis-tetrahedral).
In this class there is no centre of symmetry nor cubic planes of symmetry; the three tetrad axes become dyad axes of symmetry, and the four triad axes are polar, i.e. they are associated with different faces at their two ends. The other elements of symmetry (s; x dodecahedral planes and six dyad axes) are the same as in the last class.
Of the seven simple forms, the cube, rhombic dodecahedron and tetrakis-hexahedron are geometrically the same as before, though on actual crystals the faces will have different surface characters.
FIG. 26. - Tetrahedron. FIG. 27. - Deltoid Dodecahedron.
For instance, the cube faces will be striated parallel to only one of the diagonals (fig. 90), and etched figures on this face will be symmetrical with respect to two lines, instead of four as in the last class. The remaining simple forms have, however, only half the number of faces as the corresponding form in the last class, and are spoken of as " hemihedral with inclined faces." Tetrahedron (fig. 26). This is bounded by four equilateral triangles and is identical with the regular tetrahedron of geometry. The angles between the normals to the faces are 109° 28'. It may be derived from the octahedron by suppressing the alternate faces.
FIG. 28. - Triakis-tetrahedron. Fig. 29. - Hexakis-tetrahedron.
Deltoid 1 dodecahedron (fig. 27). This is the hemihedral form of the triakis-octahedron; it has the indices {hhk} and is bounded by twelve trapezoidal faces.
1 From the Greek letter 50vra, A; in general, a triangular-shaped object; also an alternative name for a trapezoid.
Triakis-tetrahedron (fig. 28). The hemihedral form {hkk} of the icositetrahedron; it is bounded by twelve isosceles triangles arranged in threes over the tetrahedron faces.
Hexakis-tetrahedron (fig. 29). The hemihedral form {hkl} of the hexakis-octahedron; it is bounded by twenty-four scalene triangles and is the general form of the class.
FIG. 30. - Combination of two FIG. 31. - Combination of Tetra Tetrahedra. hedron and Cube.
Corresponding to each of these hemihedral forms there is another geometrically similar form, differing, however, not only in orientation, but also in actual crystals in the characters of the faces. Thus from the octahedron there may be derived two tetrahedra with the indices {III } and { i I I}, which may be distinguished as positive and negative respectively. Fig. 30 shows a combination of FIG. 32. - Combination of FIG. 33. - Combination of Tetrahedron, Cube and Rhombic Tetrahedron and Rhombic Dodecahedron. Dodecahedron.
these two tetrahedra, and represents a crystal of blende, in which the four larger faces are dull and striated, whilst the four smaller are bright and smooth. Figs. 31-33 illustrate other tetrahedral combinations.
Tetrahedrite, blende, diamond, boracite and pharmacosiderite are substances which crystallize in this class.
Pyritohedral I Class (Parallel-faced hemihedral; Dyakis-dodecahedral). Crystals of this class possess three cubic planes of symmetry but no dodecahedral planes. There are only three dyad axes of symmetry, which coincide with the crystallographic axes; in addition there are three triad axes and a centre of symmetry.
Here the cube, octahedron, rhombic dodecahedron, triakis-octahedron and icositetrahedron are geometrically the same as in the first class. The characters of the faces will, however, be different; thus the cube faces will be striated parallel to one edge only (fig. 89), and triangular markings on the octahedron faces will be placed obliquely to the edges. The remaining simple forms are " hemihedral with parallel faces," and from the corresponding holohedral forms two hemihedral forms, a positive and a negative, may be derived.
Pentagonal dodecahedron (fig. 34). This is bounded by twelve pentagonal faces, but these are not regular pentagons, and the angles over the three sets of different edges are different. The regular dodecahedron of geometry, contained by twelve regular pentagons, is not a possible form in crystals. The indices are {hko} : as a simple form {21O}is of very common occurrence in pyrites.
Dyakis-dodecahedron (fig. 35). This is the hemihedral form of 1 Named after pyrites, which crystallizes in a typical form of this class.
FIG. 25. - Combination of FIG. 24. - Hexakis-octahedron. Hexakis - octahedron and Cube.
FIG. 34. Pentagonal Dodecahedron.
FIG. 35. Dyakis-dodecahedron.
the hexakis-octahedron and has the indices {hkl}; it is bounded by twenty-four faces. As a simple form {321 } is met with in pyrites.
Combinations (figs. 36-39) of these forms with the cube and the octahedron are common in pyrites. Fig. 37 resembles in general FIG. 36. - Combination of FIG. 37. - Combination of Pentagonal Dodecahedron Pentagonal Dodecahedron and Cube. and Octahedron.
appearance the regular icosahedron of geometry, but only eight of the faces are equilateral triangles. Cobaltite, smaltite and other sulphides and sulpharsenides of the pyrites group of minerals crystallize in these forms. The alums also belong to this class; from an aqueous solution they crystallize as simple octahedra, FIG. 38. - Combination of FIG. 39. - Combination of PentagonalDodecahedron,Cube Pentagonal Dodecahedron e and Octahedron. {210}, Dyakis-dodecahedron f {3211, and Octahedron d {III } .
sometimes with subordinate faces of the cube and rhombic dodecahedron, but from an acid solution as octahedra combined with the pentagonal dodecahedron 12 10}.
Plagihedral 1 Class (Plagihedral-hemihedral; Pentagonal icositetrahedral; Gyroidal 2).
In this class there are the full number of axes of symmetry (three tetrad, four triad and six dyad), but no planes of symmetry and no centre of symmetry.
Pentagonal icositetrahedron (fig. 40). This is the only simple form in this class which differs geometrically from those of the holosymmetric class. By suppressing either one or other set of alternate faces of the hexakis-octahedron two pentagonal icositetrahedra {hkl} and {khl} are derived. These are each bounded by twenty-four irregular FIG. 41. - Tetrahedral Pentagonal Dodecahedron.
pentagons, and although similar to each other they are respectively rightand left-handed, one being the mirror image of the other; such similar but nonsuperposable forms are said to be enantiomorphous (iv avrios, opposite, and ,uopon, form), and crystals showing such forms sometimes rotate the plane of polarization of plane-polarized light. Faces of a pentagonal icositetrahedron with high indices have been very rarely observed on crystals of cuprite, potassium chloride and ammonium chloride, but none of these are circular polarizing.
Tetartohedral Class (Tetrahedral pentagonal dodecahedral).
Here, in addition to four polar triad axes, the only other elements of symmetry are three dyad axes, which coincide with the crystallo 1 From 2rX6.ycos, placed sideways, referring to the absence of planes and centre of symmetry.
2 From yupos, a ring or spiral, and elZos, form.
graphic axes. Six of the simple forms, the cube, tetrahedron, rhombic dodecahedron, deltoid dodecahedron, triakis-tetrahedron and pentagonal dodecahedron, are geometrically the same in this class as in either the tetrahedral or pyritohedral classes. The general form is the Tetrahedral pentagonal dodecahedron (fig. 41). This is bounded by twelve irregular pentagons, and is a tetartohedral or quarter-faced form of the hexakis-octahedron. Four such forms may be derived, the indices of which are {hkl}, {khl}, {hkl} and {khl}; the first pair are enantiomorphous with respect to one another, and so are the last pair. Barium nitrate, lead nitrate, sodium chlorate and sodium bromate crystallize in this class, as also do the minerals ullmannite (NiSbS) and langbeinite (K2Mg2(S04)3).
2. Tetragonal SYSTEr (Pyramidal; Quadratic; Dimetric).
In this system the three crystallographic axes are all at right angles, but while two are equal in length and interchangeable the third is of a different length. The unequal axis is spoken of as the principal axis or morphological axis of the crystal, and it is always placed in a vertical position; in five of the seven classes of this system it coincides with the single tetrad axis of symmetry.
The parameters are a: a: c, where a refers to the two, equal hori FIG. 42.
Tetragonal Bipyramids.
zontal axes, and c to the vertical axis; c may be either shorter (as in fig. 42) or longer (fig. 43) than a. The ratio a: c is spoken of as the axial ratio of a crystal, and it is dependent on the angles between the faces. In all crystals of the same substance this ratio is constant, and is characteristic of the substance; for other substances crystallizing in the tetragonal system it will be different. For example, in cassiterite it is given as a: c= I: 0.67232 or simply as c =0.67232, a being unity; and in anatase as c = 1 7771.
Holosymmetric Class (Holohedral; Ditetragonal bipyramidal).
Crystals of this class are symmetrical with respect to five planes, which are of three kinds; one is perpendicular to the principal axis, and the other four intersect in it; of the latter, two are perpendicular to the equal crystallographic axes, while the two others bisect the angles between them. There are five axes of symmetry, one tetrad and two pairs of dyad, each perpendicular to a plane of symmetry. Finally, there is a centre of symmetry.
There are seven kinds of simple forms, viz.: Tetragonal bipyramid of the first order (figs. 42 and 43). This is bounded by eight equal isosceles triangles. Equal lengths are intercepted on the two horizontal axes, and the indices are {III }, 12211, 11121, &c., or in general {hhl}. The parametral plane with the intercepts a: a: c is a face of the bipyramid 11111.
Tetragonal bipyramid of the second order. This is also bounded by eight equal isosceles triangles, but differs from the last form in FIG. 44. FIG. 45. Tetragonal Bipyramids of the first and second orders.
its position, four of the faces being parallel to each of the horizontal axes; the indices are therefore {101}, {201}, {102}, &c., or Pon .
Fig. 44 shows the relation between the tetragonal bipyramids FIG. 40. - Pentagonal Icositetrahedron.
FIG. 43.
of the first and second orders when the indices are {III} and {IoI} respectively: ABB is the face (III), and ACC is (tot). A combination of these two forms is shown in fig. 45.
Ditetragonal bipyramid (fig. 46). This is the general form; it is bounded by sixteen scalene triangles, and all the indices are unequal, being 13211, &c., or {hkl} Tetragonal prism of the first order. The four faces intersect the horizontal axes in equal lengths and are parallel to the principal axis; the indices are therefore {IIo}. This form does not enclose space, and is therefore called an " open form " to distinguish it from a " closed form " like the tetragonal bipyramids and all the forms of the cubic system. An open form can exist only in combination with other forms; thus fig. 47 is a combination of the tetragonal prism {IIo} with the basal pinacoid tom }. If the faces (Ho) and (001) are of equal size such a figure will be geometrically a cube, since all the angles are right angles; the variety of apophyllite known as tesselite crystallizes in this form.
Tetragonal prism of the second order.
FIG. 46. - Ditetragonal This has the same number of faces as Bipyramid. the last prism, but differs in position; each face being parallel to the vertical axis and one of the horizontal axes; the indices are {loo}.
Ditetragonal prism. This consists of eight faces all parallel to the principal axis and intercepting the horizontal axes in different lengths; the indices are 12'0, {320}, &c., or {hko}. Basal pinacoid (from 7rivaE, a tablet). This consists of a single pair of parallel faces perpendicular to the principal axis. It is therefore an open form and can exist only in combination (fig. 47).
Combinations of holohedral tetragonal forms are shown in figs. 47-49; fig. 48 is a combination of a bipyramid of the first order with one of the second order and the prism of the first order; fig. 49 a same way that two tetrahedra are derived from the regular octahedron.
Tetragonal scalenohedron or ditetragonal bisphenoid (fig. 51). This is bounded by eight scalene triangles and has the indices {hkl} It may be considered as the hemihedral form of the ditetragonal bipyramid.
FIG. 50. - Tetragonal Bisphenoids. FIG. 51. - Tetragonal Scalenohedron.
The crystal of chalcopyrite (CuFeS 2) represented in fig. 52 is a combination of two bisphenoids (P and P'), two bipyramids of the second order (b and c), and the basal pinacoid (a). Stannite (Cu2FeSnS4), acid potassium phosphate (H2KP04), mercuric cyanide, and urea (CO(NH 2) 2) also crystallize in this class.
Bipyramidal Class (Parallel-faced hemihedral).
| FIG. 47. Combination of Tetragonal Prism and Basal Pinacoid. | ||||||
|---|---|---|---|---|---|---|
The elements of symmetry are a tetrad axis with a plane perpendicular to it, and a centre of symmetry. The simple forms are the same here as in the holosymmetric class, except the prism {hko}, which has only four faces, and the bipyramid {hkl}, which has eight faces and is distinguished as a " tetragonal pyramid of the third order." FIG. 52. - Crystal of Chalcopyrite. FIG. 53. - Crystal of Fergusonite.
FIG. 48. FIG. 49. Combinations of Tetragonal Prisms and Pyramids.
combination of a bipyramid of the first order with a ditetragonal bipyramid and the prism of the second order. Compare also figs. 87 and 88.
Examples of substances which crystallize in this class are cassiterite, rutile, anatase, zircon, thorite, vesuvianite, apophyllite, phosgenite, also boron, tin, mercuric iodide.
Scalenohedral Class (Bisphenoidal-hemihedral).
Here there are only three dyad axes and two planes of symmetry, the former coinciding with the crystallographic axes and the latter bisecting the angles between the horizontal pair. The dyad axis of symmetry, which in this class coincides with the principal axis of the crystal, has certain of the characters of a tetrad axis, and is sometimes called a tetrad axis of " alternating symmetry "; a face on the upper half of the crystal if rotated through 90° about this axis and reflected across the equatorial plane falls into the position of a face on the lower half of the crystal. This kind of symmetry, with simultaneous rotation about an axis and reflection across a plane, is also called " composite symmetry." In this class all except two of the simple forms are geometrically the same as in the holosymmetric class.
Bisphenoid (vcPi i v, a wedge) (fig. 50). This is a double wedgeshaped solid bounded by four equal isosceles triangles; it has the indices {III }, 12111, 11121, &c., or in general {hhl} . By suppressing either one or other set of alternate faces of the tetragonal bipyramid of the first order (fig. 42) two bisphenoids are derived, in the Fig. 53 shows a combination of a tetragonal prism of the first order with a tetragonal bipyramid of the third order and the basal pinacoid, and represents a crystal of fergusonite. Scheelite, scapolite, and erythrite (C4H1004) also crystallize in this class.
Pyramidal Class (Hemimorphic-tetartohedral).
Here the only element of symmetry is the tetrad axis. The pyramids of the first {hhl} , second {ho/} and third {hkl} orders have each only four faces at one or other end of the crystal, and are hemimorphic. All the simple forms are thus open forms.
Examples are wulfenite (PbM004) and barium antimonyl dextrotartrate (Ba(Sb0)2(C4H406).H20).
Ditetragonal Pyramidal Class (Hemimorphic-hemihedral).
Here there are two pairs of vertical planes of symmetry intersecting in the tetrad axis. The pyramids {hhl} and {ho/} and the bipyramid {hkl} are all hemimorphic.
Examples are iodosuccimide (C 4 H 4 0 2 NI), silver fluoride (AgF.H20), and penta-erythrite (C6H1204). No examples are known amongst minerals.
Trapezohedral Class (Trapezohedral-hemihedral).
Here there are the full number of axes of symmetry, but no planes or centre of symmetry. The general form {hkl} is bounded by eight trapezoidal faces and is the tetragonal trapezohedron.
vim 19 Examples are nickel sulphate (NiSO 4.6H 2 0), guanidine carbonate ((CH 5 N 3) 2 H 2 CO 3), strychnine sulphate((C21H22N202)2 H2S04.6H20).
Bisphenoidal Class (B isphenoidal-tetartohedral) .
Here there is only a single dyad axis of symmetry, which coincides with the principal axis. All the forms, except the prisms and basal pinacoid, are sphenoids. Crystals possessing this type of symmetry have not yet been observed.
(Rhombic; Prismatic; Trimetric).
In this system the three crystallographic axes are all at right angles, but they are of different lengths and not interchangeable. The parameters, or axial ratios, are a: b: c, these referring to the axes OX, OY and OZ respectively. The choice of a vertical axis, OZ = c, is arbitrary, and it is customary to place the longer of the two horizontal axes from left to right (OY=b) and take it as unity: this is called the " macro-axis " or " macro-diagonal " (from µaKpos, long), whilst the shorter horizontal axis (OX =a) is called the " brachy-axis " or " brachy-diagonal " (from 1 3pax6s, short). The axial ratios are constant for crystals of any one substance and are characteristic of it; for example, in barytes (BaS04), a: b: c= 0.8152: I: 1.3136; in anglesite (PbS04), a: b: c = 0.7852: I: 1 2894; in cerussite (PbC03), a:b:c= 0.6100: 1 :0.7230.
There are three symmetry-classes in this system Holohedral Class (Holohedral; Bipyramidal).
Here there are three dissimilar dyad axes of symmetry, each coinciding with a crystallographic axis; perpendicular to them are three dissimilar planes of symmetry; there is also a centre of symmetry. There are seven kinds of simple forms: Bipyramid (figs. 54 and 55). This is the general form and is bounded by eight scalene triangles; the indices are {III}, {21I}, FIG. 55. Orthorhombic Bipyramids.
{221}, {112}, {321}, {123}, &c., or in general {hkl}. The crystallographic axes join opposite corners of these pyramids and in the fundamental bipyramid {iii} the parametral plane has the intercepts a: b: c. This is the only closed form in this class; the others are open forms and can exist only in combination. Sulphur often crystallizes in simple bipyramids.
Prism. This consists of four faces parallel to the vertical axis and intercepting the horizontal axes in the lengths a and b or in any multiples of these; the indices are therefore II io}, 12101, { 120} or {hko}.
Macro-prism. This consists of four faces parallel to the macro FIG. 57. - Brachy-prism and Macro-pinacoid.
axis, and has the indices { I o I }, 12011. or {ho/1.
Brachy-prism. This consists of four faces parallel to the brachyaxis, and has the indices OW I }, 10211. .. {okl}. The macroand brachy-prisms are often called " domes." Basal pinacoid, consisting of a pair of parallel faces perpendicular to the vertical axis; the indices are loos}. The macro-pinacoid {loo} and the brachy-pinacoid {010} each consist of a pair of parallel faces respectively parallel to the macroand the brachy-axis.
Figs. 56-58 show combinations of these six open forms, and fig. 59 a combination of the macro-pinacoid (a), brachy-pinacoid (b), a prism (na), a macro-prism (d), a brachy-prism (k),and a bipyramid (u). FIG. 58. - Prism and Basal FIG. 59. - Crystal of Pinacoid. Hypersthene. Holohedral Orthorhombic Combinations.
Examples of substances crystallizing in this class are extremely numerous; amongst minerals are sulphur, stibnite, cerussite, chrysoberyl, topaz, olivine, nitre, barytes, columbite and many others; and amongst artificial products iodine, potassium permanganate, potassium sulphate, benzene, barium formate, &c.
Pyramidal Class (Hemimorphic).
Here there is only one dyad axis in which two planes of symmetry intersect. The crystals are usually so placed that the dyad axis coincides with the vertical crystallographic axis, and the planes of symmetry are also vertical.
The pyramid {hkl} has only four faces at one end or other of the crystal. The macro-prism and the brachy-prism of the last class are here represented by the macro-dome and brachy-dome respectively, so called because of the resemblance of the pair of equally sloped faces to the roof of a house. The form {ooi } is a single plane at the top of the crystal, and is called a "pedion "; the parallel pedion tool}, if present at the lower end of the crystal, constitutes a different form. The prisms {hko} and the macroand brachy-pinacoids are geometrically the same in this class as in the last. Crystals of this class are therefore differently developed at the two ends and are said to be " hemimorphic." Fig. 60 shows a crystal of the mineral hemimorphite (H2Zn2S105) which is a combination of the brachy-pinacoid {010} and a prism, FIG. 60. - Crystal of FIG. 61. - Orthorhombic Hemimorphite. Bisphenoid.
with the pedion (ooi), two brachy-domes and two macro-domes at the upper end, and a pyramid at the lower end. Examples of other substances belonging to this class are struvite (NH 4 MgPO 4.6H 2 O), bertrandite (H2Be4S1209), resorcin, and picric acid.
Bisphenoidal Class (Hemihedral).
Here there are three dyad axes, but no planes of symmetry and no centre of symmetry. The general form thk11 is a bisphenoid (fig. 61) bounded by four scalene triangles. The other simple forms are geometrically the same as in the holosymmetric class.
Examples: epsomite (Epsom salts, MgSO 4.7H 2 O), goslarite (ZnSO 4.7H 2 0), silver nitrate, sodium potassium dextro-tartrate (seignette salt, NaKC 4 H 4 O 6.4H 2 0), potassium antimonyl dextrotartrate (tartar-emetic, K(SbO)C 4 H 4 0 6), and asparagine (C4H8N208.H20).
FIG. 54.
FIG. 56. - Macro-prism and Brachy-pinacoid.
4. Monoclinic i System (Oblique; Monosymmetric).
In this system two of the angles between the crystallographic axes are right angles, but the third angle is oblique, and the axes are of unequal lengths. The axis which is perpendicular to the other two is taken as OY=b (fig. 62) and is called the ortho-axis or orthodiagonal. The choice of the other two axes is arbitrary; the vertical axis (OZ =c) is usually taken parallel to the edges of a prominently developed prismatic zone, and the clino-axis or clino-diagonal (OX =a) parallel to the zone-axis of some other prominent zone on the crystal. The acute angle between the axes OX and OZ is usually denoted as /3, and it is necessary to know its magnitude, in addition to the axial ratios a: b: c, before the crystal is completely determined. As in other systems, except the cubic, these elements, a: b: c and /3, are characteristic of the substance. Thus for gypsum a: b: c =o 6899: I:0 4124; /3=80° 42'; for orthoclase a: b: c= 0.6585: I: 0.5554; 1 3=63° 57'; and for cane-sugar a: b: c = 1 2 595 I: 0.8782; 0 = 76° 76° 30'.
Holosymmetric Class (Holohedral; Prismatic).
Here there is a single plane of symmetry perpendicular to which is a dyad axis; there is also a centre of symmetry. The dyad axis coincides with the ortho-axis 0 Y, and the vertical axis OZ and the clino-axis OX lie in the plane of symmetry.
All the forms are open, being either pinacoids or prisms; the former consisting of a pair of parallel faces, and the latter of four faces intersecting in parallel edges and with a rhombic cross-section. The pair of faces parallel to the plane of symmetry is distinguished as the " clino-pinacoid " and has the indices 10101. The other pinacoids are all perpendicular to the plane of symmetry (and parallel to the ortho-axis); the one parallel to the vertical axis is called the " ortho-pinacoid " 11001, whilst that parallel to the clinoaxis is the " basal pinacoid " {001}; pinacoids not parallel to the arbitrarily chosen clinoand vertical axes may have the indices 11011, 12011, {1021. .. (hol) or {IoI}, {201}, {102}. .. {viol}, according to whether they lie in the obtuse or the acute axial angle. Of the prisms, those with edges (zone-axis) parallel to the clino-axis, and having indices 10T1, 1021}, {012}. .. {okl}, are called " clinoprisms "; those with edges parallel to the vertical axis, and with the indices {11O}, {210}, 11201. .. {hko}, are called simply " prisms." Prisms with edges parallel to neither of the axes OX and OY have the indices 11II}, 12211, 12111, {3211. .. {hkl} or {III}. .. {hkl}, FIG. 62. - Monoclinic Axes and Hemi-pyramid.
and are usually called "hemi :pyramids " (fig. 62); they are distinguished as negative or positive according to whether they lie in the obtuse or the acute axial angle /3.
Fig. 63 represents a crystal of augite bounded by the clinopinacoid (1), the ortho-pinacoid (r), a prism (M), and a hemi-pyramid (s).
The substances which crystallize in this class are extremely numerous: amongst minerals are gypsum, orthoclase, the amphiboles, pyroxenes and micas, epidote, monazite, realgar, borax, mirabilite (Na 2 SO 4.10H 2 0), melanterite (FeSO 4.7H 2 O) and many others; amongst artificial products are monoclinic sulphur, barium chloride (BaC1 2.2H 2 O), potassium chlorate, potassium ferrocyanide (K 4 Fe(CN)e 3H 2 O), oxalic acid (C 2 0 4 H 2.2H 2 0), sodium acetate (NaC 2 H 3 O 2.3H 2 O) and naphthalene.
Hemimorphic Class (Sphenoidal).
In this class the only element of symmetry is a single dyad axis, which is polar in character, being dissimilar at the two ends.
The form 10101 perpendicular to the axis of symmetry consists of a single plane or pedion; the parallel face is dissimilar in character and belongs to the pedion 10701. The pinacoids {loo}, {ooi}, {hol} and {hol} parallel to the axis of symmetry are geometrically the From ,uovos, single, and KXivecv, to incline, since one axis is inclined to the plane of the other two axes, which are at right angles.
same in this class as in the holosymmetric class. The remaining forms consist each of only two planes on the same side of the axial plane XOZ and equally_ inclined to the dyad axis (e.g. in fig. 62 the two planes XYZ and X YZ); such a wedge-shaped form is sometimes called a sphenoid.
FIG. 64. - Enantiomorphous Crystals of Tartaric Acid.
Fig. 64 shows two crystals of tartaric acid, a a right-handed crystal of dextro-tartaric acid, and b a left-handed crystal of laevotartaric acid. The two crystals are enantiomorphous, i.e. although they have the same interfacial angles they are not superposable, one being the mirror image of the other. Other examples are potassium dextro-tartrate, cane-sugar, milk-sugar, quercite, lithium sulphate (L12S04 H20); amongst minerals the only example is the hydrocarbon fichtelite (C5H8).
Clinohedral Class (Hemihedral; Domatic).
Crystals of this class are symmetrical only with respect to a single plane. The only form which is here geometrically the same as in the holosymmetric class is the clino-pinacoid {oio}. The forms perpendicular to the plane of symmetry are all pedions, consisting of single planes with the indices (Ioo), (Too), (ooi), (ooi), (hol), &c. The remaining forms, {hko}, (okl) and (hkl), are domes or " gonioids " (ycovia, an angle, and ei os, form), consisting of two planes equally inclined to the plane of symmetry.
Examples are potassium tetrathionate (K 2 S 4 0 6), hydrogen trisodium hypophosphate (HNa3P206.9H20); and amongst minerals, clinohedrite (H 2 ZnCaSiO 4) and scolectite.
5. Anorthic System (Triclinic).
In the anorthic (from Its, privative, and 6p00s, right) or triclinic system none of the three crystallographic axes are at right angles, and they are all of unequal lengths. In addition to the parameters a: b: c, it is necessary to know the angles, a, /3, and -y, between the axes. In anorthite, for example, these elements are a: b: c= o.6 347: 1 :o 5501; a =93° 13', #=115° 55', y = 91° 12'.
Holosymmetric Class (Holohedral; Pinacoidal).
Here there is only a centre of symmetry. All the forms are pinacoids, each consisting of only two parallel faces. The indices of the three pinacoids parallel to the axial planes are {ioo}, {oio} and 10011; those of pinacoids parallel to only one axis are {hko}, {hol} and {okl}; and the general form is {hkl}. Several minerals crystallize in this class; for example, the plagioclastic felspars, microcline, axinite (fig. 65), cyanite, amblygonite, chalcanthite(CuSO 4.5H 2 O),sassolite(H 3 B0 3); among artificial substances are potassium bichromate, racemic acid (C4H606.2H20), dibrom-para-nitrophenol, &c.
Asymmetric Class (Hemihedral, Pediad).
Crystals of this class are devoid of any elements of symmetry. All the forms are pedions, each consisting of a single plane; they are thus hemihedral with respect to crystals of the last class. Although there is a total absence of symmetry, yet the faces are arranged in zones on the crystals.
Examples are calcium thiosulphate (CaS20 3.6H 2 0) and hydrogen strontium dextro-tartrate ((C4H40cH) 2 Sr 5H 2 O); there is no example amongst minerals.
6. Hexagonal System Crystals of this system are characterized by the presence of a single axis of either triad or hexad symmetry, which is spoken of as the " principal " or " morphological " axis. Those with a triad axis are grouped together in the rhombohedral or trigonal division, and those with a hexad axis in the hexagonal division. By some authors these two divisions are treated as separate systems; or again the rhombohedral forms may be considered as hemihedral developments FIG. 63. - Crystal of Augite.
FIG. 65. - Crystal of Axinite.
of the hexagonal. On the other hand, hexagonal forms may be considered as a combination of two rhombohedral forms.
Owing to the peculiarities of symmetry associated with a single triad or hexad axis, the crystallographic axes of reference are different in this system from those used in the five other systems of crystals. Two methods of axial representation are in common use; rhombohedral axes being usually used for crystals of the rhombohedral division, and hexagonal axes for those of the hexagonal division; though sometimes either one or the other set is employed in both divisions.
Rhombohedral axes are taken parallel to the three sets of edges of a rhombohedron (fig. 66). They are inclined to one another at equal oblique angles, and they are all equally inclined to the principal axis; further, they are all of equal length and are interchangeable. With such a set of axes there can be no statement of an axial ratio, but the angle between the axes (or some other angle which may be calculated from this) may be given as a constant of the substance. Thus in calcite the rhombohedral angle (the angle between two faces of the fundamental rhombohedron) is 74° 55', or the angle between the normal to a face of this rhombohedron and the principal axis is 44° 362'.
Hexagonal axes are four in number, viz. a vertical axis coinciding with the principal axis of the crystal, and three horizontal axes inclined to one another at 60° in a plane perpendicular to the principal axis. The three horizontal axes, which are taken either parallel or perpendicular to the faces of a hexagonal prism (fig. 71) or the edge of a hexagonal bipyramid (fig. 70), are equal in length (a) but the vertical axis is of a different length (c). The indices of planes referred to such a set of axes are four in number; they are written as {hikl}, the first three (h+i+k = o) referring to the horizontal axes and the last to the vertical axis. The ratio a: c of the parameters, or the axial ratio, is characteristic of all the cr y stals of the same substance. Thus for beryl (including emerald) a: c 0.4989 (often written c =0.4989); for zinc c =1.3564 In the rhombohedral or trigonal division of the hexagonal system there are seven symmetry-classes, all of which possess a single triad axis of symmetry.
Holosymmetric Class (Holohedral; Ditrigonal scalenohedral).
In this class, which presents the commonest type of symmetry of the hexagonal system, the triad axis is associated with three similar planes of symmetry inclined to one another at 60° and inter FIG. 66. FIG. 67. Direct and Inverse Rhomboliedra.
secting in the triad axis; there are also three similar dyad axes, each perpendicular to a plane of symmetry, and a centre of symmetry. The seven simple forms are: Rhombohedron (figs. 66 and 67), consisting of six rhomb-shaped faces with the edges all of equal lengths: the faces are perpendicular to the planes of symmetry. There are two sets of rhombohedra, distinguished respectively as direct and inverse; those of one set (fig. 66) are brought into the orientation of the other set (fig. 67) by a rotation of 60° or 180° about the principal axis. For the fundamental rhombohedron, parallel to the edges of which are the crystallographic axes of reference, the indices are 1'001. Other rhombohedra may have the indices 12111, 14111, {IIo}, 12211, {III}, &c., or in general {hkk}. (Compare fig. 72; for figures of other rhombohedra see Calcite.) Scalenohedron (fig. 68), bounded by twelve scalene triangles, and with the general indices {hkl}. The zig-zag lateral edges coincide with the similar edges of a rhombohedron, as shown in fig. 69; if the indices of the inscribed rhombohedron be Imo}, the indices of the scalenohedron represented in the figure are {2(31 1.. The scalenohedron 12011 is a characteristic form of calcite, which for this reason is sometimes called " dog-tooth-spar." The angles over the three edges of a face of a scalenohedron are all different; the angles over three alternate polar edges are more obtuse than over the other three polar edges. Like the two sets of rhombohedra, there are also direct and inverse scalenohedra, which may be similar in form and angles, but different in orientation and indices.
| FIG. 70. - Hexagonal Bipyramid. | ||
| FIG. 69. - Scalenohedron with inscribed Rhombohedron. | FIG. 71. - Hexagonal Prism and Basal Pinacoid. |
Hexagonal bipyramid (fig. 70), bounded by twelve isosceles triangles each of which are equally inclined to two planes of symmetry. The indices are {210}, {41a}, &c., or in general (hkl), where h-2k+1 Hexagonal prism of the first order (211), consisting of six faces parallel to the principal axis and perpendicular to the planes of symmetry; the angles between (the normals to) the faces are 60°.
Hexagonal prism of the second order (101), consisting of six faces parallel to the principal axis and parallel o o the planes of symmetry. The faces of this prism are inclined to 30 to those of the last prism.
Dihexagonal prism, consisting of twelve faces parallel to the principal axis and inclined to the planes of symmetry. There are two sets of angles between the faces. The indices are 13211, {532}. {hkl}, where h+k+l=o. Basal pinacoid 1111), consisting of a pair of parallel faces perpendicular to the principal axis.
Fig. 71 shows a combination of a hexagonal prism (m) with the basal pinacoid (c). For figures of other combinations see Calcite i12 //2 FIG. 72. - Stereographic Projection of a Holosymmetric Rhombohedral Crystal.
and Corundum. The relation between rhombohedral forms and their indices are best studied with the aid of a stereographic projection (fig. 72); in this figure the thicker lines are the projections of the three planes of symmetry, and on these lie the poles of the rhombohedra (six of which are indicated).
Numerous substances, both natural and artificial, crystallize 110 FIG. 68. - Scalenohedron.
in this class; for example, calcite, chalybite, calamine, corundum (ruby and sapphire), haematite, chabazite; the elements arsenic, antimony, bismuth, selenium, tellurium and perhaps graphite; also ice, sodium nitrate, thymol, &c.
Ditrigonal Pyramidal Class (Hemimorphic-hemihedral) .
Here there are three similar planes of symmetry intersecting in the triad axis; there are no dyad axes and no centre of symmetry. The triad axis is uniterminal and polar, and the crystals are differently developed at the two ends; crystals of this class are therefore pyro-electric. The forms are all open forms: Trigonal pyramid {hkk}, consisting of three faces which correspond to the three upper or the three lower faces of a rhombohedron of the holosymmetric class.
Ditrigonal pyramid {hkl}, of six faces, corresponding to the six upper or lower faces of the scalenohedron.
Hexagonal pyramid (hkl) (where h-2k+ 1= o), of six faces, corresponding to the six upper or lower faces of the hexagonal bipyramid.
Trigonal prism PHI } or {21 I }, two forms each consisting of three faces parallel to principal axis and perpendicular to the planes of symmetry.
Hexagonal prism { Ioi }, which is geometrically the same as in the last class.
Ditrigonal prism !hkl} (where h+k+l= o), of six faces parallel to the principal axis, and with two sets of angles between them.
Basal pedion (III) or (iii), each consisting of a single plane perpendicular to the principal axis.
Fig. 73 represents a crystal of tourmaline with the trigonal prism (211"), hexagonal prism (Ioi), and a trigonal pyramid at each end. Other substances crystallizing in this class are pyrargyrite, proustite, iodyrite (AgI), greenockite, zincite, spangolite, sodium lithium sulphate, tolylphenylketone.
Trapezohedral Class (Trapezohedral-hemihedral).
Here there are three similar dyad axes inclined to one another at 60° and perpendicular to the triad axis. There are no planes or centre of symmetry. The dyad axes are uniterminal, and are pyroelectric axes. Crystals of most substances of this class rotate the plane of polarization of a beam of light.
In this class the rhombohedra {hkk}, the hexagonal prism {2 i i }, and the basal pinacoid /III} are geometrically the same as in the holosymmetric class; the trigonal prism {loll and the ditrigonal prisms are as in the ditrigonal pyramidal class. The remaining simple forms are: Trigonal trapezohedron (fig. 74), bounded by six trapezoidal faces. There are two complementary and enantiomorphous trapezohedra, {hkl} and {hlk}, derivable from the scalenohedron.
Trigonal bipyramid (fig. 75), bounded by six isosceles triangles; the indices are {hkl}, where h-2k+1=o, as in the hexagonal bipyramid.
The only minerals crystallizing in this class are quartz (q.v.) and cinnabar, both of which rotate the plane of a beam of polarized light transmitted along the triad axis. Other examples are dithionates of lead (PbS 2 0 6.4H 2 0), calcium and strontium, and of potassium (K 2 S 2 0 6), benzil, matico-stearoptene.
Rhombohedral Class (Parallel-faced hemihedral).
The only elements of symmetry are the triad axis and a centre of symmetry. The general form {hkl} is a rhombohedron, and is a hemihedral form, with parallel faces, of the scalenohedron. The form {Well , where h-2k+l = o, is also a rhombohedron, being the hemihedral form of the hexagonal bipyramid. The dihexagonal prism {hkl} of the holosymmetric class becomes here a hexagonal prism. The rhombohedra (hkk), hexagonal prisms 1211 - 1 and {1 oi}, and the basal pinacoid /1 I 1 } are geometrically the same in this class as in the holosymmetric class.
Fig. 76 represents a crystal of dioptase with the fundamental rhombohedron r loo} and the hexagonal prism of the second order ?n {Ioi} combined with the rhombohedron s log).
Examples of minerals which crystallize in this class are phenacite, dioptase, willemite, dolomite, ilmenite and pyrophanite: amongst artificial substances is ammonium periodate ((NH4)41209.3H20).
Trigonal Pyramidal Class (Hemimorphic-tetartohedral).
Here there is only the triad axis of symmetry, which is uniterminal. The general form {hkl} is a trigonal pyramid consisting of three faces at one end of the crystal. All other forms, in which the faces are neither parallel nor perpendicular to the triad axis, are trigonal pyramids. All the prisms are trigonal prisms; and perpendicular to these are two pedions.
The only substance known to crystallize in this class is sodium periodate (Na104.3H20), the crystals of which are circularly polarizing.
Trigonal Bipyramidal Class Here there is a plane of symmetry perpendicular to the triad axis. The trigonal pyramids of the last class are here trigonal bipyramids (fig. 75); the prisms are all trigonal prisms, and parallel to the plane of symmetry is the basal pinacoid. No example is known for this class.
Dioptase.
Here there are three similar planes of sym metry intersecting in the triad axis, and perpendicular to them is a fourth plane of symmetry; at the intersection of the three vertical planes with the horizontal plane are three similar dyad axes; there is no centre of symmetry.
The general form is bounded by twelve scalene triangles and is a ditrigonal bipyramid. Like the general form of the last class, this has two sets of indices {hkl, pqF}, (hkl) for faces above the equatorial plane of symmetry and (p F) for faces below: with hexagonal axes there would be only one set of indices. The hexagonal bipyramids, the hexagonal prism {Ioi } and the basal pinacoid l I I I } are geometrically the same in this class as in the holosymmetric class. The trigonal prism pH} i } and ditrigonal prisms {hki} are the same as in the ditrigonal pyramidal class.
The only representative of this type of symmetry is the mineral benitoite.
In crystals of this division of the hexagonal system the principal axis is a hexad axis of symmetry. Hexagonal axes of reference are used: if rhombohedral axes be used many of the simple forms will have two sets of indices.
Holosymmetric Class (Holohedral; Dihexagonal bipyramidal).
Intersecting in the hexad axis are six planes of symmetry of two kinds, and perpendicular to them is an equatorial plane of symmetry. Perpendicular to the hexad axis are six dyad axes of two kinds and each perpendicular to a vertical plane of symmetry. The seven simple forms are: Dihexagonal bipyramid, bounded by twenty-four scalene triangles (fig. 77; v in fig. 80). The indices are {2131 }, &c., or in general {hikl} . This form may be considered as a combination of two scalenohedra, a direct and an inverse.
Hexagonal bipyramid of the first order, bounded by twelve FIG. 78. FIG. 79. FIG. 80. Combinations of Hexagonal forms.
isosceles triangles (fig. 70; p and u in fig. 80); indices {Io11}, {2021 }. .. (hohl). The hexagonal bipyramid so common in quartz is geometrically similar to this form, but it really is a combination of two rhombohedra, a direct and an inverse, the faces of which differ in surface characters and often also in size.
a m a a FIG. 73. - Crystal of Tourmaline.
FIG. 74. - Trigonal FIG. 75. - Trigonal Trapezohedron. Bipyramid.
Ditrigonal Bipyramidal Class Fig. 76. - Crystal of Bipyramid.
Hexagonal bipyramid of the second order, bounded by twelve faces (s in figs. 79 and 80); indices { I 121 }, 111221.. {h. h.2 h.l }. Dihexagonal prism, consisting of twelve faces parallel to the hexad axis and inclined to the vertical planes of symmetry; indices {hiko}. Hexagonal prism of the first order {ioio}, consisting of six faces parallel to the hexad axis and perpendicular to one set of three vertical planes of symmetry (m in figs. 71, 78-80).
Hexagonal prism of the second order {I120}, consisting of six faces also parallel to the hexad axis, but perpendicular to the other set of three vertical planes of symmetry (a in fig. 78).
Basal pinacoid {0001}, consisting of a pair of parallel planes perpendicular to the hexad axis (c in figs. 71, 78-80).
Beryl (emerald), connellite, zinc, magnesium and beryllium crystallize in this class.
Bipyramidal Class (Parallel-faced hemihedral).
Here there is a plane of symmetry perpendicular to the hexad axis; there is also a centre of symmetry. All the closed forms are hexagonal bipyramids; the open forms are hexagonal prisms or the basal pinacoid. The general form {hikl} is hemihedral with parallel faces with respect to the general form of the holosymmetric class.
Apatite, pyromorphite, mimetite and vanadinite possess this degree of symmetry.
Dihexagonal Pyramidal Class (Hemimorphic-hemihedral).
Six planes of symmetry of two kinds intersect in the hexad axis. The hexad axis is uniterminal and all the forms are open forms. The general form lhikl} consists of twelve faces at one end of the crystal, and is a dihexagonal pyramid. The hexagonal pyramids {hold} and (h.h.2h.l) each consist of six faces at one end of the crystal. The prisms are geometrically the same as in the holosymmetric class. Perpendicular to the hexad axis are the pedions (0001) and (0001).
Iodyrite (AgI), greenockite (CdS), wurtzite (ZnS) and zincite (ZnO) are often placed in this class, but they more probably belong to the hemimorphic-hemihedral class of the rhombohedral division of this system.
Trapezohedral Class (Trapezohedral-hemihedral).
Six dyad axes of two kinds are perpendicular to the hexad axis. The general form Ihik11 is the hexagonal trapezohedron bounded by twelve trapezoidal faces. The other simple forms are geometrically the same as in the holosymmetric class. Barium-antimonyl dextro-tart rate-}-potassium nitrate(Ba (SbO) 2 (C 4H 4 0 6) 2. KNO 3) and the corresponding lead salt crystallize in this class.
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 sometimes be grouped in parallel position and so give rise to special structures, of which the dendritic (from bEvSpov, 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. Miigge, " Die regelmassigen Verwachsungen von Mineralien verschiedener Art," Neues Jahrbuck filr Mineralogie, 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 a plane 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 (ioo), 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 180 0 about the normal to the same plane. Such a crystal (fig. 81) is therefore described as being twinned on the plane (100).
An octahedron (fig. 8 3) twinned on an octahedral face (11I) 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 compositionplane, 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 " interpenetration 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 180° about a triad axis, or by reflection across the octahedral plane which is perpendicular to this axis; the twinplane is therefore (III).
Since in many cases twinned crystals may be explained by the rotation of one portion through two right angles, R. J. Rally introduced the term " hemitrope " (from the Gr. iv-, half, and T poiros, a turn); the word " made " 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 FIG. 84. - Interpenetrating Twinned Cubes.
by reflection across a plane; these are known as " symmetric twins," a good example of which is furnished by one of the twinlaws 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, FIG. 83. - Spinel-twin.
FIG. 81. - Twinned FIG. 82. - Simple Crystal of Gypsum. Crystal of Gypsum.
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), Glideplane). 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 iog° 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. Although from a geometrical point of view these figures are no FIG. 86. 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 1 6 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 FIG. 87. - Actual Crystal. FIG. 88. - Ideal Development.
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 su