EARTH FENCES IBRARY B ,?KELEY LIBRARY UNIVERSITY OF CALIFORNIA LIBRARY ELEMENTARY CRYSTALLOGRAPHY Published by the McGraw-Hill Boot Company Succe>s.sons to tneBooK-Departments or the McGraw Publishing Company Hill Publishing Company Fxiblishers of Book* "for Electrical World The Engineering" and Mining Journal The Engineering Record R>wer and The Engineer Electric Railway Journal American Machinist ELEMENTARY CRYSTALLOGRAPHY BEING PART ONE OF GENERAL MINERALOGY BY W. S. BAYLEY, Ph. D. ASSOCIATE PROFESSOR OF MINERALOGY AND ECONOMIC GEOLOGY UNIVERSITY OF ILLINOIS. McGRAW-HILL BOOK COMPANY 239 WEST 39TH STREET, NEW YORK 6 BOUVERIE STREET, LONDON, E. C. 1910 COPYRIGHT, 1910 BY THE McGRAW-HiLL BOOK COMPANY Printed and Electrotyped by The Maple. Press York, Pa. PREFACE. The material in the following pages was originally intended to be the introductory portion of a text-book on Mineralogy. Because of the increasing interest in Crystallography on the part of chemists and others, however, it has been finally decided to offer it as an independent volume, in the hope that it might find a wider use in this form than if it were merely a portion of a larger volume which would appeal mainly to mineralogists. It is intended that it shall be followed in the near future by a second volume which will deal with minerals the two taken together being expected to constitute an elementary text-book on Mineralogy. The present volume does not pretend to be a treatise on Crystallography. It is merely a guide for those attempting to gain some insight into the fundamental principles underlying the science. It is the direct result of the need felt by the author in his own classes for a simple discussion of crystals and a series of simple statements of crystallographic conceptions. It is not intended to be a description of crystal forms, nor an illustrated list of crystallographic symbols. It is hoped that it may serve as an aid in teaching, but is not expected to serve as a reference book. Consequently some of the facts usually contained in text-books on Crystallography will not be found in its pages. Whether the material retained has always been chosen with wisdom must be left to the judgment of its readers. It was written for students of Mineralogy, Chemistry, and Physics. The mineralogist has always used crystallography as an effi- cient aid in his study of minerals. The chemist is beginning to appreciate its value in his work. The physicist will before long discover that it is absolutely necessary to employ its methods 819467 VI PREFACE in studying the physical properties of many of the substances with which he has to deal before he may hope to reach com- parable results. Because of the use to which the book is to be put, the author has assumed but little preliminary knowledge of mathematics, physics, and chemistry on the part of its users. Technical terms have been employed as rarely as possible, consistent with clear- ness, and only those topics are discussed which are necessary to the understanding of the nature of crystals. The discussion of optical phenomena has been reduced to a minimum, because of their complex nature, and the difficulty of observing them without a supply of expensive apparatus. The choice of the Naumann System of parameter symbols in preference to the more elegant Miller System of indices has been made with deliberateness, solely because of the greater ease with which the Naumann symbols are comprehended by students who approach the study of crystals for the first time. The indices, however, are indicated for all the forms described or pictured. There is no claim that any portion of the volume is original, except perhaps its method of developing the discussion. The illustrations have been collected from various sources. A few are original. Others have been taken from the text-books of Williams, Groth arid Linck. When possible to do so, their sources have been indicated. The writer is under obligations to Dr. R. M. Bagg, Jr., for photographs of some of the minerals illustrated in the text and to the publishers for their painstaking efforts to give the volume an attractive dress. CONTENTS. Introduction. Mineralogy: Its Object, History, and Division into Branches . IX CHAPTER CHAPTER I. II. CHAPTER CHAPTER CHAPTER III. IV. V. CHAPTER VI. CHAPTER VII. CHAPTER VIII, CHAPTER CHAPTER IX. X. CHAPTER XL CHAPTER XII. CHAPTER XIII. CHAPTER XIV. CHAPTER XV. PART I. GEOMETRICAL CRYSTALLOGRAPHY. General Facts of Crystallography ..... 3 The Law of the Constancy of Interfacial Angles 14 The Law of Simple Mathematical Ratios . . 17 Symmetry 25 The Isometric, or Regular, System Holo- hedral Division 31 Partial Forms Hemihedrism and Tetartohe- drism of the Isometric System 41 The Hexagonal System 54 Holohedral Division 60 Hemihedral Division 69 Tetartohedral Division 82 The Tetragonal System 87 Holohedral Division 87 Hemihedral Division 95 The Orthorhombic System 102 Holohedral Division 105 Hemihedral Division . 112 The Monoclinic System 115 Holohedral Division . 117 Hemihedral Division 121 The Triclinic System 125 Crystal Imperfections 132 Crystal Aggregates 143 Amorphous Substances and Pseudomorphs . 160 Crystal Projection .164 vii VI 11 CONTENTS. PART II. PHYSICAL CRYSTALLOGRAPHY. CHAPTER XVI. Introduction: Physical Agencies and Phys- CHAPTER XVII. CHAPTER XVIII. CHAPTER XIX. CHAPTER XX. ical Symmetry ' . 177 Mechanical Properties of Crystals 180 Optical Properties of Crystals . . . . . . .196 Thermal Properties of Crystals 209 Electrical and Magnetic Properties of Crystals . 214 PART III. CHEMICAL CRYSTALLOGRAPHY. CHAPTER XXI. CHAPTER XXII. Solution and Etched Figures. . Isomorphism and Polymorphism 219 227 INTRODUCTION. MINERALOGY: ITS OBJECT, HISTORY, AND DIVISIONS INTO BRANCHES. The Purpose of Mineralogy is the study of the solid, homo- geneous, inorganic compounds occurring in nature, either as the constituents of rock masses or as the components of veins, which are the fillings of chinks or fissures in the earth's crust. The science is closely related to Chemistry, since the objects of its study are definite chemical compounds, all of whose proper- ties are believed to be determined by their composition, and to Geology, because their aggregations make up by far the greater portion of the earth's crust the rocks. The principles made use of in the study of minerals are not peculiar to the science of Mineralogy. Since minerals are definite chemical compounds they must be studied in part by chemical methods. The properties by which they are known are partly geometrical and partly physical, and these must be investigated by mathematical and physical processes. Miner- alogy is thus not a pure science with methods of study peculiar to itself. It is a mixed science. Its problems are attacked by methods that are borrowed from other sciences and applied to minerals for the purpose of discovering their nature. Distinction between Minerals and Rocks. A mineral is any definite inorganic compound, either solid or liquid, occur- ring in nature. It is not the direct result of life processes nor the product of man's experiments. A rock is a definite portion of the earth's crust irrespective of its composition. It is usually a mass of, sufficient size to be regarded as an architectural unit in the structure known as the earth's crust. It does not necessarily possess a definite chemical composition, nor is it usually homogeneous. It may be an aggre- ix X INTRODUCTION gate of minerals of the same kind, as limestone, which is composed of grains of the mineral calcite (CaCO 3 ), or it may be a mixture of minerals of different kinds, as granite, which consists es- sentially of feldspar, (a potassium-aluminium silicate), and quartz (SiO 2 ), or it may be an accumulation of organic substances, like coal, or a mass of undifferentiated glass. Different portions of the same rock when analyzed will usually give different results, and in only the few rare instances where a single mineral may constitute its entire mass can its composition be represented by a simple chemical formula. According to the definitions given, water (H 2 O) is a mineral, as are also quartz (SiO 2 ) magnetic iron ore (Fe 3 O 4 ), diamond (C), etc. Coal, amber, and wood are not minerals, because they are the products of organic processes. Granite and obsidian (volcanic glass) are not minerals because they do not possess definite chemical compositions. Tin is not a mineral because it is the product of man's work, while natural tin oxide, the ore from which the metal is obtained, is the mineral cassiterite or tin stone. The copper that occurs as native copper is a mineral, while the copper obtained by smelting the sulphide is an artificial product. History of Mineralogy. Although the study of the metals and of gems dates back certainly to the time of Aristotle (384 B. C.), the scientific study of the less valuable minerals began with Haiiy (1743), a famous French teacher, who discovered that all minerals possess characteristic forms which may be deduced in some cases from their physical properties. In the last half of the eighteenth century and in the early years of the nineteenth century, Werner (1750) and Weiss (1808), in Germany, described in great detail a large number of minerals, and the latter author developed a method of classifying and representing their forms. From this time until the present day Germans have been the most earnest students of Mineralogy in all its branches. France, Great Britain, and Italy have many devoted mineralogists, while North America has produced quite a number who have acquired world-wide reputations. The science has recently become so broad that no one pretends INTRODUCTION XI to be versed in all its branches. There are descriptive miner- alogists, physical mineralogists, optical mineralogists, mathe- matical mineralogists, and chemical mineralogists, the members of each class devoting themselves to special phases of the science, but all being more or less thoroughly acquainted with the results of the work in all its other phases. Divisions of the Subject. Since Mineralogy is such a broad science, dealing as it does with the geometrical, physical, and chemical properties of minerals, it has been found convenient to divide it into branches, each one of which is devoted to the study of some special class of properties. The fundamental basis of mineralogy is chemistry. All minerals are definite chemical compounds, the properties of each one of which are directly dependent upon their chemical com- position and structure. Thus Chemical Mineralogy is one branch of the subject, and upon it all the other branches are founded. When different chemical compounds are studied it is soon discovered that they vary in color, hardness, ease of fusibility, etc., i.e., in their physical properties. We therefore recognize a branch of mineralogy that is devoted to the investigation of the physical properties of minerals Physical Mineralogy. Again, careful scrutiny of these same compounds soon reveals the fact that individual minerals are characterized by definite and distinctive forms by which they may be recognized. Thus a piece of quartz (SiO 2 ) may often be distinguished from a piece of calcite (CaCO 3 ) simply by its external form. Morpho- logical Mineralogy deals with the forms of minerals. But distinctive forms are not confined to minerals. They are pos- sessed likewise by other substances which have been produced under conditions analogous to those under which minerals are formed. They are characteristic of nearly all solid substances that have been sublimed from gases or vapors or have been precipitated from solutions. The study of such forms from the general point^ of view is known as Crystallography. Further, bodies exhibiting these definite geometrical forms are also characterized by certain well-defined physical properties which Xll INTRODUCTION are closely related to the forms exhibited. Consequently, Crystal lography is naturally divisible into two branches, viz., Geo- metrical Crystallography and Physical Crystallography. When minerals are studied with respect to their individual properties, we have Descriptive Mineralogy. After becoming acquainted with the characteristics of different minerals it often becomes necessary to make use of these charac- teristics in determining the nature of an unknown substance. This application of our knowledge concerning the properties of known minerals to the discovery of the nature of an unknown mineral is called Determinative Mineralogy. In Chemistry it corresponds to Analytical Chemistry. The present volume deals only with the elements of Geo- metrical and Physical Crystallography. A later volume will be devoted to a discussion of the general characteristics of minerals and descriptions of the features of the most common kinds. PART I. GEOMETRICAL CRYSTALLOGRAPHY. CHAPTER I. GENERAL FACTS OF CRYSTALLOGRAPHY. Forms Assumed by Substances. The forms assumed by substances through the agency of natural forces are of two classes : (i) those produced by the action of internal molecular forces and (2) those produced by external agencies. The first depend upon the nature of the material composing the substance, and are known as Idiomorphic forms; the second are to a large extent independent of the nature of the substance, but are determined by the nature of the mechanical agents acting upon it and the directions along which they act, or by the methods by which it was produced. They are accidental forms. Since the forms FIG. i. Crystals of calcite attached at one end. FIG. 2. Stalactite of calcite. of this class are not directly dependent upon the nature 'of the bodies exhibiting them, they cannot be classified and studied systematically. Figure i illustrates an idiomorphic form (a group of crystals of calcite (CaCO 3 )), and figure 2, an accidental form (a stalactite of the same substance). The shape of the crystals (Fig. i) is due to the fact that they are crystals of calcite. Crystals of other sub- stances have not the same shape as this one. The shape of the stalactite (Fig. 2) is due to the fact that the calcite of which it is 3 4* GEOMETRICAL CRYSTALLOGRAPHY composed was deposited from dripping water and not to the fact that its substance is calcite. There are stalactites of many sub- stances not at all similar to calcite. Among other forms of the second class may be mentioned those of mineral pebbles, frag- ments, cut stone, like the sets in rings, etc., all of which were produced by forces which had their origin outside the substance. Idiomorphic Forms. On the contrary, the idiomorphic forms are directly dependent upon the nature of the substance of which they are characteristic. Calcite (CaCO 3 ) always possesses definite forms that are distinctive for this mineral ; that is, exactly the same form is never seen in other minerals. Quartz (SiO 2 ) also has its distinctive forms, which are peculiar to itself. So with many other substances. Occasionally several substances which are apparently iden- tical in chemical composition may have different idiomorphic forms, as is the case with calcium carbonate (CaCO 3 ). The apparent identity of such substances is, however, believed not to be a real identity. There are two calcium carbonates which differ from one another in the structure of their chemical molecule or in the way in which their molecules are united. They are, therefore, different chemical compounds. These calcium car- bonates possess different idiomorphic forms, as we should expect of them, and, therefore, they are given different names. One of them is the mineral calcite and the other the mineral aragonite. Idiomorphic forms may be defined as those which are original to the body on which they occur, and which are essential to its substance. Molecules and Crystal Particles. All bodies are believed to be made up of tiny particles called molecules, which vary in size and weight for different substances. They are so very small that the most powerful microscope will not detect them. In a single cubic inch of a cold gas like nitrogen they are present to the number of about no billion billions. These molecules are so small that it would take about 55,000,000 to make a row one inch long. Nevertheless, if all thaT are present in a cubic inch of gas were placed in a row, touching one another, they would form a line 32,000,000 miles long. The molecules are always in motion, GENERAL FACTS OF CRYSTALLOGRAPHY 5 but they are supposed not to touch each other even in the hardest and densest bodies. They are separated from one another by tiny spaces. When a body passes from the liquid or gaseous state to the solid form, the little molecules tend to arrange themselves into definite groupings, which are the crystal particles. These are built up by the addition of other particles until they become large enough to be seen first by the microscope, then by the naked eye, until finally they reach a size that may be measured by inches or even feet. All the little particles of a body built u in this way are constructed on the same plan, so that every portion of the body has the same structure. If we should take bits from its different parts and could magnify them sufficiently to render their component molecules visible, we would find these little bodies all arranged in a definite manner, which would be the same for every fragment. If we could examine other bodies in the same way, we should find that they, too, are composed of molecules arranged in certain definite groupings; but, in many cases, we should discover these groupings to be different from those of the first body studied. While we cannot see the method of grouping of the particles, we can infer something of its character from the phenomena presented by different kinds of bodies. From the effects produced we can study the groupings themselves, and thus can gain an insight into the internal structure of the bodies which they compose. From a careful consideration of the subject from all points of view it is concluded that the arrangement of the molecules in any solidifying body is dependent almost exclusively upon the nature of the little molecules themselves. It is believed that the power to build up well-defined and characteristic groupings is a property of the molecules, just as the power of chemical combination between the atoms is an inherent property of the atoms. But the chemical composition of every homogeneous body is the same as the chemical composition of its component molecules, hence we may infer that the form assumed by a body is dependent upon its composition. 6 GEOMETRICAL CRYSTALLOGRAPHY In order to explain the peculiarities of a body in its different states it seems necessary to assume that every molecule possesses attractive powers by which it tends to pull toward itself other molecules of the same kind, and that at the same time it is endowed with the impulse to move in straight lines. These two properties which are supposed to be inherent in every molecule tend to counteract one another in part. When the tendency to move away from its neighbors (the repulsive tendency) predominates in the molecules composing a body, that body is said to be a gas. When the attractive tendency overcomes the repulsive one, the body is a liquid or a solid. The distinction between the solid and the liquid states depends upon other considerations such, perhaps, as the friction between their constituent molecules. The two pass gradually into one another so that it is often diffi- cult to discriminate between them. A body is usually said to be a liquid if it assumes a spherical form when floating in a medium with which it does not mix. Crystalline Bodies and Crystals. Not all solid substances are constructed of regularly arranged molecules. In some the molecules possess no regularity of arrangement, so far as we can learn. This class includes such bodies as glasses, jellies, etc., which are spoken of as amorphous or colloidal. Other substances, like the tissues of plants and animals, are constructed of cells regularly built up, but this regularity is not the result of the piling up of the molecules in a definite manner in consequence of any property inherent in them, but is the result of the life processes of the animals or plants that make the cells. The molecules in the cells are not regularly and definitely arranged, although the shapes of the little cells themselves may all be alike. In other words, while the tissue is composed of regularly arranged cells, the materials composing the cells may be amorphous. When a solid or fluid body is composed throughout of molecules arranged according to some definite plan, it is said to be crystalline; when composed of molecules arranged helter-skelter, i.e., without any definite plan, it is said to be amorphous or colloidal. The distinction between an amorphous body of definite form and a crystalline body may be easily shown with the aid of a few GENERAL FACTS OF CRYSTALLOGRAPHY 7 buckshot. If a layer of these be built into a triangle, as illustrated in figure 3 a, and on this layer a second layer be placed in the manner indicated, and upon this a third layer, and so on, we finally obtain a pile of shot constructed of regularly arranged components (Fig. 3 b). The shape of the pile depends upon the plan of the foundation triangle and upon the manner in which the successive layers are piled upon the first one. If we let the shot represent molecules, the pile represents a crystalline body with a regular internal structure. On the other hand, we may tumble the shot indiscriminately into several little boxes all made exactly alike. The masses of shot within the boxes would all have the same shapes the shape of the boxes but there would be no regularity in the arrangement of the shot. The contents of the boxes illustrate the structure of an amorphous body. Since crystalline bodies exhibit their physical properties in different degrees along different directions, a second assumption with regard to molecules is necessary, viz., that their activities are oriented. In other words, the lines along which their activities are exerted are definitely arranged with respect to one another. When we are dealing with an amorphous body in which there is no regularity in the arrangement of its molecules, there is, as a con- sequence, no regularity in the arrangement of the directions along which their powers act; consequently such bodies possess similar properties in parallel directions. Thus a sphere of glass, which 8 GEOMETRICAL CRYSTALLOGRAPHY is amorphous, remains a sphere under all conditions of tempera- ture, i.e., it expands equally in all directions. On the other hand, a sphere of quartz, which is crystalline, loses its spherical form when the temperature rises or falls and becomes ellipsoidal, i.e., it expands and contracts to a greater degree in one direction than in others. Its physical properties exhibit a definite arrangement. Crystals. When the little molecules of crystalline bodies are given the opportunity and the time to arrange themselves in accordance with the plan that suits them best, they build up structures usually bounded by planes, which vary in number and in inclination to one another according to certain definite laws. A homogeneous crystalline body thus bounded by planes is called a crystal, or a crystal individual. Or, since the bounding planes are the result of the internal structure, a crystal may be defined as a homogeneous body bounded by a polyhedron that is idiomorphic. The term homogeneous is introduced into the definition because it sometimes happens that a single body bounded by plane faces consists of several parts united in such a way that, while the composition and arrangement of the molecules in all portions of each part is the same, the composition and arrange- ment in the different parts is different. The term homogeneous signifies that all parts of the individual are like all other parts. Although the great majority of crystalline bodies are solid, there are some liquids which under proper conditions exhibit a definite internal structure which is recognizable in polarized light (see page 196) and which, when in the form of drops, show a tendency to bound themselves by plane surfaces. These are known as liquid crystals. They do not possess the sharply defined forms of solid crystals, but are distinctly bounded by surfaces that are nearly plane and that intersect in edges (see Fig. 4). Crystallization. The process by which crystals and crystal- line bodies are formed is known as crystallization. No satis- factory explanation of the force of crystallization has yet been proposed. It is known that some substances possess this power to a much greater degree than others, and in some (the colloids) it appears to be entirely lacking. In the latter cases, however, GENERAL FACTS OF CRYSTALLOGRAPHY 9 the power may still exist, but if so it is so weak that it can express itself only under the most favorable conditions. Crystallography. The external forms of crystallized bodies are their most striking peculiarities, consequently they have been studied longer than their other features. They were at first supposed to be their most essential characteristics, and hence their study was early named crystallography. It is, however, now well recognized that the forms are only one of the methods through which the molecular structure of a crystallized body expresses itself. The term as now used includes not only the FIG. 4. Photograph of liquid crystals with rounded edges. (After O. Lehmann.) study of the forms of crystals, but also the study of all their other characteristic properties. Sometimes the former is dis- tinguished as geometrical crystallography and the latter as physical and chemical crystallography. Crystallography includes the study of the forms assumed by all crystallized bodies, whether these bodies occur as minerals or whether they exist only as the products of the laboratory. When applied to mineral substances it is often called ''Morphological Mineralogy." Laws of Crystallography. Almost as soon as the forms of crystals began to be studied carefully it was discovered that the planes which comprise them are fixed in position and that their relations to one another are so well-defined that they may be ex- pressed mathematically. The crystallographic laws thus dis- covered are very simple and are the foundation of all modern crystallography. They are three in number, as follows : i. The law of the constancy of the interfacial angles on all crystals of the same substance. 10 GEOMETRICAL CRYSTALLOGRAPHY 2. The law of the rationality of the indices; or the law of sim- ple mathematical ratios. 3. The law of symmetry. Definitions. Before attempting to discuss the meanings of the laws and their significance it is necessary to become acquainted with some of the commonest terms used in describing crys- tals. These are not many in number nor are they difficult of comprehension. An interfacial angle is the angle included between any two of the planes or faces of a crystal or between their prolongations (ABC in Fig. 5). FIG. 6. Zo al arrangement of planes on crystal. A crystal angle is the solid angle in which three or more faces meet (D, E in Fig. 5). A crystal edge or interfacial edge is the line in which two con- tiguDus crystal faces meet (E-E, D-D in Fig. 5). A zone of planes is a belt of planes whose edges with each other are parallel lines (see Fig. 6). A zonal axis is a line passing through the center of a crystal parallel to the edges of a zone of planes. All the planes lying in a zone are parallel to the zonal axis for that zone (see Fig. 6) . The planes c,o,o 2 ,m,o 2 ,o in front, constitute one zone on the crystal illustrated in figure 6, o,q,s,m,s,q form another, and the planes GENERAL FACTS OF CRYSTALLOGRAPHY II marked m a third. The zonal axis of the third zone is the line AB, that of the first zone is CD, and that of the second zone EF, provided the lines are regarded as passing through the center of the crystal.* Measurement of Interfacial Angles. In order to determine the elements of crystals it is necessary that the vajfcies of their interfacial angles should be known, as the inclination of the plajie? to one another varies with the values of these angles, or, to put FIG. 7. Contact goniometer. FIG. 8. Simple form of con- tact goniometer. the case more logically, the values of the interfacial angles vary with the inclination of the planes including them. The instruments with which the interfacial angles are meas- ured are known as goniometers. The simplest form is the contact goniometer (Fig. 7). This form consists of a graduated arc and two detachable arms, one of which revolves about a pivot common to both. These arms are removed from the arc and applied to the two faces of the crystal whose angles are to be measured, care being taken to hold them perpendicular to the interfacial edge, and at the same time to press them firmly * Before entering further upon the study of crystals the student should familiar- ize himself with the terms denned, by applying them to the proper parts of a few simple crystals or crystal models. First the edges, the interfacial and the crystal angles should be pointed out, and then all the zones occurring on a few more com- plicated models should be determined. A very useful and cheap set of 60 crystal models in wood may be obtained from dealers in minerals. Other and more comprehensive sets may be obtained from Dr. F. Krantz, Bonn on Rhine, Germany, at an average cost of 45 cents for each model. 12 GEOMETRICAL CRYSTALLOGRAPHY against the faces. The screw holding the arms is then set, and the arms themselves are replaced on the arc, when the value of the angle desired is indicated by the graduation. Measurements made with the contact goniometer are reliable to about half a degree. Other forms of the contact goniometer are even simpler than the one described. Their construction, however, is so similar to that of the one illustrated that they demand no description (see Fig. 8). A more accurate but much more complicated instrument is the reflection goniometer (Fig. 9). Its use depends upon the prin- FIG. 9. Reflection goniometer. ciple that a ray of light falling upon two planes in parallel posi- tion is reflected from them in parallel directions. The mechan- ism of the instrument is such as to give a fixed ray of light (through L in the figure), a means of detecting its reflection (O), and an arrangement for so placing a crystal that its con- tiguous planes may be brought successively into a given position (K). By the use of a well-made instrument the angle through which the crystal must be turned in order that its faces shall give GENERAL FACTS OF CRYSTALLOGRAPHY 13 parallel reflections can be read off to within a fraction of a minute of arc on the graduated circle V. This angle is the supplement of the interfacial angle. Let AOC (in Fig. 10) be the interfacial angle to be measured and EP a ray of light falling upon the face AO. It will be re- flected along the line PG. In order that the same ray shall be reflected from the face CO in the same direction this face must be brought into the position OC', when OA will have the position OA'. But in order to bring OC into the position OC' the crystal must be turned through the angle COC', which is the supplement of the interfacial angle COA. Importance of Accurate Measurements of Interfacial Angles. The importance of de- termining accurately the values of interfacial angles is realized FIG when we remember that only those planes that are parallel to each other can be equally inclined to a given plane, hence planes that make different angles with a given plane cannot be parallel to one another. Moreover, since in crystallography planes are distinguished from each other by their inclinations to standard planes, it follows that planes that make different angles with a given plane are different. It further fol- lows that when two planes on one crystal have a different inter- facial angle than that possessed by two planes on a second crystal, the planes on the two crystals cannot be the same. The values of interfacial angles may thus serve to distinguish between substances that are otherwise apparently identical. For instance, certain light Jm>wn crystals of calcite (CaCO 3 ) bear a strong resemblance to certain crystals of light brown siderite (FeCO 3 ). If the crystals are six-sided (rhombohedrons) they contain corresponding interfacial angles, which on the calcite have a value of 106 15', and on the siderite 107. By careful measurement this difference may be recognized and the two min- erals may be distinguished. CHAPTER II. THE LAW OF THE CONSTANCY OF INTERFACIAL ANGLES. Statement of the Law. The observation and comparison of crystals have shown that however much the crystals of a given substance may vary in size and shape, their corresponding interfacial angles are the same in value, provided they be measured under the same conditions. Significance of the Law. This law expresses the fact that it is not the general shape, or habit, of a crystal that is its most important external feature, but the inclination of the planes by which it is bounded. Two crystals may look very much alike, i.e., they may be similar in habit, but if the inclinations of their FIG. ii. Three crystals bounded by the same planes, but with different habits. corresponding faces are different the crystals themselves are made up of different crystallographic elements. On the other hand, if two crystals that are apparently entirely unlike each other are bounded by planes with their corresponding interfacial angles equal in value, the corresponding crystal faces are regarded as the same faces and the crystals are crystallographically identical. The crystals A,B, and C represented in the subjoined figure (Fig. u) are very different in habit, and yet they are bounded 14 THE LAW OF THE CONSTANCY OF INTERFACIAL ANGLES 15 by the same planes, since the interfacial angles on A are identical in value with the corresponding angles on B and C (between planes marked with the same letters). The condition imposed by the law, viz., that the crystals shall "be measured under the same conditions," is necessitated by the fact that crystals deport themselves toward physical agencies in a different manner from non-crystallized substances. Under the influence of changes in temperature, for instance, they may expand or contract differently in different directions, and the inclinations of their faces will vary accordingly. Even the same crystal will yield different values for the same interfacial angle when this is measured at different temperatures, hence in comparing the angles on different crystals it is necessary to know at what temperature they were measured in order to decide whether or not the faces including the angles are the same. The expression of the law further indicates that it applies only to crystals of the same substance. From this we may rightly infer that crystals of different substances possess different interfacial angles even though bounded by similar crystallographic planes. As a fact, except among the bodies crystallizing in what is known as the regular or isometric system, and with the exception of a very few angles on other crystallized bodies, the crystals of different substances possess different interfacial angles, which, however, are so distinctive for each substance that they may often serve as a means for its identification. The values of the interfacial angles are therefore characteristic for different substances. They must consequently vary according as the chemical composition of the substance varies. When the direction and amount of the variation caused in the value of any angle by the admixture of a certain chemical element into the molecule is known the quantity of this element introduced may frequently be determined by the difference in the value of the l6 GEOMETRICAL CRYSTALLOGRAPHY angle observed and that of the corresponding angle in the pure substance (see pages 231-232). In the mineral calcite (CaCO 3 ) the angle between the faces R and R' is 74 55' (Fig. 12), while in magnesite (MgCO 3 ), which crystallizes like calcite, it is 72 31'. In the mineral dolomite (MgCaCO 3 ), which crystallizes in the same shape, the corre- sponding angle is 73 45'. This is between that of calcite and that of magnesite. Morphotropism. The influence which the introduction of an element or a group of elements into a compound, or their abstraction from a compound, exerts upon the values of the interfacial angles is known as morphotropism. The action itself is known as morphotropic action. The morphotropic action of Mg in calcite, for instance, is to decrease the value of the angle RAR'. Corollary to the Law. Since the interfacial angles are the important elements in defining the nature of crystals, and not the sizes or shapes of the faces or their distances from the centers of the crystals, and since the only planes that can make equal interfacial angles with a given plane are those that are parallel to each other, it follows that parallel faces in crystallography may be regarded as the same face, situated at different distances from the center of the crystal. Thus a small cube and a large cube are crystallographically identical forms, which differ from each other only in the distances of their bounding planes from the centers of the respective forms. CHAPTER III. THE LAW OF SIMPLE MATHEMATICAL RATIOS. The Designation of the Positions of Planes in Space. The position of a plane in space may be denned by referring it to a system of three lines intersecting at a common point, just as the position of a line on a plane may be defined by referring it to two other intersecting lines. Planes on crystals are referred to a system of imaginary lines, known as axes, which are supposed to intersect at the center of the crystal. The positions and inclinations of the planes are -b- -fb FIG. 13. System of axes. FIG. 14. defined by expressing the relations between the distances at which they cut the axes, measured from the point of their common intersection. In figure 13 is illustrated a scheme of axes to which the planes on certain crystals may be referred. In order to locate the planes accurately each axis is designated by a letter, and the two ends of each axis are distinguished by different signs. The positions of planes are thus easily indicated by stating the relative distances at which they cut the three axes. The plane ABC (Fig. 14) may be defined as a plane cutting the a axis at the distance OA, the b 17 1 8 GEOMETRICAL CRYSTALLOGRAPHY I axis at OB, and the c axis at OC. The plane HKL may likewise be defined as a plane cutting a at OH, b at OK, and c at OL. Parameters. In all measurements of distance a standard length is assumed, in terms of which the result of the measurement is expressed. In ordinary practice the standards made use of are the inch, foot, yard, mile, or metre, kilometre, etc. In crystallography the standard lengths are those distances at which " a selected plane, called the groundplane or groundform, cuts, or intercepts, the axes. These distances are taken as units, and the distances at which other planes on the same crystal intercept the corresponding axes are expressed in terms of these units. - The resulting expressions are known as the parameters of the plane. If ABC in figure 14 is taken as a groundplane, then its intercepts OA, OB, and OC on the three axes are the unit lengths on these axes. The plane HKL intercepts the axes at OH, OK, and OL. If OH- 2 OA; OK = OB, and OL == 1/2 OC, then the position of HKL may be expressed by stating that its intercept on a is twice the unit length on a, on b the intercept is the unit on b, and on c the intercept is 1/2 the unit on c, or, briefly, the symbol representing the plane is 2a : b : i/2C, in which a, b, and c repre- sent the unit lengths on the three axes. The values 2, i, and 1/2 are the parameters of the plane on the axes a, b, c, respectively. General Symbol for a Plane. The most general symbol for any plane is ma : nb : pc, in which m, n, and p represent definite values other than one. A plane with this symbol cuts the three axes at different distances, and no one of the distances is the unity for that axis. But parallel planes in crystallography are regarded as the same plane (see page 16), hence one may imagine them moved parallel to themselves without affecting their relations to other planes. The plane ma, nb, pc, may be shifted in position m p until it cuts b at unity. It will then cut a at a, and c at -c, pro- vided its new position is parallel to its original one, and the symbol of the plane will become m'a : b : p'c. The parameters are thus ratios between the intercepts of a plane on the axes and the unity distances on the same axes, when the plane is moved paral- lel to itself until it intercepts one of the axes at a unity (or THE LAW OF SIMPLE MATHEMATICAL RATIOS standard) distance. By this method corresponding planes of similar crystals are represented by the same symbol whether the crystal be small or large. The parameters are ratios and not absolute distances. One of the theorems of solid geometry states that if two paral- lel planes are cut by any third plane, the intersections are parallel. Let ABC (Fig. 15) be a plane cutting the axes a, b, and c at A, B, and C, respectively, and let OA = w, O~B=n, and OC = #. Let A', B', C' be the position of this plane when it is moved parallel to itself until it cuts b at unity (B 7 ). Pass a plane through the axes OC' (c) and OB 7 (6), then will the intersections of this plane with FIG. 15. ABC and A'B'C', or the lines BC and B'C', be parallel. The triangles OBC and OB'C' are therefore similar, and OB : OB' : : OC : OC'=n : i : : p : OC', and OC' (the intercept of A'B'C' on c) =5-. In a similar way it can be proven that the new inter- n m cept on a = . n The values m, n, and p of the first symbol are not the same as m', i, and p' of the second symbol, but the ratios between m, n, and p are the same as those between m f , i , and p'. This being the case, it is convenient to express the symbol of a plane in the form of a ratio, for then it becomes a general symbol for any plane parallel to a given plane. Thus m'a : b : p'c is the symbol for any plane cutting the three axes in the ratio m' times, once, and p' times the unity distances on a, b, and c. For instance, the symbol 2a : b : i/2C designates any plane that cuts the axes at distances that are in the ratio of 2 : i : 1/2 times the unities on the three axes. The symbols 40 : 2b : c and a : i/2b : 1/40 designate the same plane as does the symbol 20 GEOMETRICAL CRYSTALLOGRAPHY 2a : b : i/2C because the ratios between the parameters are the same in all cases. The plane 40 : 2b : c may be regarded as be- ing at twice the distance from the center of the crystal as is the plane 2a : b : i/2C, and the plane a : 1/26 : 1/40 at only half this distance; but all the planes make the same interfacial angles with the corresponding adjacent planes and hence are regarded as the same plane. If the three symbols express any difference in the planes it is simply with respect to their actual distance from the center of the crystals on which they are found; or, in other words, they express simply a difference in the size of the crystals. A cube is a cube, whether it measures i/io of an inch or 10 inches on an edge. Crystallography takes no account of the size of crystals; it deals only with the distribution and inclination of the planes that compose them. Crystallographic Notation. In discussing crystals and crystal planes it is necessary to make use of some method of representing them accurately and at the same time graphically. The principles upon which one system of notation is based are explained in the previous paragraph. This is known as the system of parameters or the Weiss system, after its inventor. The Weiss System of Notation. In this system the crys- tallographic axes are invariably written in the order a : b : c, and to these are prefixed the parameters, when these vary from unity. The symbol a : b : c without any parameters always refers to a plane of the groundform, that is, to a plane cutting the three axes at the distances assumed as the unities. The paral- lelism of a plane to a crystallographic axis is indicated by the sign of infinity, co , written in its proper place as a parameter. Thus oca : b : c, represents a plane^arallel to the a axis and cutting b and c at unity, oca : co& : c, is a plane cutting c at unity, and at the same time parallel to a and b. Further, it is customary for simplicity's sake to make the parameter on either the a or the b axis unity, and to reduce the symbol accordingly. For instance, if the measurement of a crystal should yield 420 : 6b : I2C as the symbol for a certain plane, this would be reduced to the form 70 : b : 2C, which would become the symbol for the plane. By reference to figure 13 (page 17) it will be noted that this THE LAW OF SIMPLE MATHEMATICAL RATIOS 21 symbol stands for a plane in the upper right-hand division or oc- tant of space included within the halves of the three axes. Corre- sponding planes in the other octants on the upper half of the crystal would be represented by 'ja : b : 20; ja: b:2C, and 7a : b : ooc (parameters) ooP c** (Naumann symbol) * In the Naumann symbols the signs ~ and refer to the a axis and the sign - to the b axis. When no sign is used above the parameter after the initial letter the parameter refers to the a and the 6 axes indifferently. CHAPTER IV. SYMMETRY. Symmetry. By the symmetry of a body is meant the regular arrangement of its parts with respect to planes, lines, or points. Four types of symmetry are recognizable in crystals, of which three are much more important than the fourth. These three are: 1. Symmetry with reference to planes. 2. Symmetry with respect to lines. 3. Symmetry with respect to points. Symmetry with Respect to Planes.-^ When two halves of a body bear to each other the relation of an object and its image, the two halves are said to be symmetrical about the plane that divides them. When the body possesses only one such plane of symmetry, it is described as being symmetrical about a single plane. (See Fig. 16.) When it possesses three such planes, it is symmetrical about three planes. The planes are known as planes of symmetry. Symmetry with Respect to Lines. Many bodies may be revolved about a line so situated within it that a revolution through an FIG. 16. Body angle less than 360 Will Cause its parts to divided by one plane occupy successively the same relative positions of s y mmetr y- as they originally occupied. When such a condition is brought about by a revolution through 180, or twice during a complete revolution, the body is said to possess a twofold or binary sym- metry. When brought about by revolutions of 120, 90, or 60, it is spoken of as possessing threefold or ternary, fourfold or quadratic, and sixfold or hexagonal symmetry. The line about which such a revolution is possible is called an axis of symmetry. The axes of twofold symmetry are designated as secondary 2 5 26 GEOMETRICAL CRYSTALLOGRAPHY axes of symmetry; those of more than twofold symmetry are designated as principal axes of symmetry. Symmetry axes are polar when their opposite ends terminate differently; i.e., in crystals, when their opposite ends terminate in different planes, edges, or solid angles (compare Fig. 33). Symmetry with Reference to Points. Bodies are sym- metrical about a point when a straight line drawn through this point terminates at equal distances on its opposite sides in similar surfaces or angles. Such a point is called a center of symmetry. A crystal possessing a center of symmetry must be bounded by pairs of parallel planes (see Figs. 200 and 201). A sphere and a cube are symmetrical about a point at their centers. A right cone with a circular base and a right pyramid with a square base are symmetrical about lines passing through their apices and the centers of their bases. The latter possesses a fourfold symmetry and has four vertical planes of symmetry, two passing through the corners of the base and two through the centers of its sides. A right pyramid with a rectangular base that is not a square possesses two vertical planes of symmetry through the centers of the sides of its base. A right pyramid with an isosceles triangle as its base possesses a single plane of symmetry passing vertically through its base and bisecting the angle between its equal sides. Symmetry in Crystallography. It has already been re- marked that the inclinations of the planes on a crystal are of much more importance than their distances from its center. Similarly the notion of symmetry in crystals applies to the directions of its planes rather than to the distances of these planes from the plane of symmetry. Crystallographically the two figures on the next page (Fig. 17) are symmetrical about the plane AB passed through them per- pendicular to the plane of the paper. The two parts of figure 17 A are symmetrical not only with respect to the inclinations of their bounding lines, but also with respect to the distances of these lines from the plane of symmetry. This polygon is ideal in its sym- metry. Figure 17 B is symmetrical with respect to the in- clination of the lines on opposite sides of the plane AB, but SYMMETRY not with respect to the distances of these lines from the plane. Every line on either side of the plane of symmetry corresponds to a line on the opposite side, and the corresponding lines on each side make equal angles with the corresponding adjacent lines. The angles a, b, c are equal, respectively, to a', b', c'. u A B FIG. 17. Diagrams illustrating symmetry. In the same way the crystals B and C in figure 18 are sym- metrical in the same degree as is the crystal A. But the latter crystal is ideal in its symmetry, while the other two are distorted through irregularities in their growth. Crystal Forms. One definition of a crystal form has been FIG. 18. Three crystals bounded by the same planes, but with different habits. given (page 21). It may better be defined as the sum of all the planes demanded by a crystal's symmetry, in consequence of the presence of one. It is for this reason that the symbol of a single plane of a form may be made to stand for the entire form. Occasionally through accidents of growth one or more planes of a form may be crowded from the crystal. Plane z in the upper 28 GEOMETRICAL CRYSTALLOGRAPHY left-hand corner of crystal C, figure 18, for instance, is very small. Continued growth of P and r might have caused it to disappear completely. An accident of this kind is not recognized in the symbol of a crystal, though the absence of the plane is usually mentioned in the description of the crystal's habit. Grades of Symmetry. Differences in the number of planes, axes, and centers of symmetry possessed by crystal forms deter- mine their grade of symmetry the greater the number of elements of symmetry present, the higher the grade of symmetry. Law of Symmetry. Each crystallizing substance possesses a characteristic grade of symmetry. Only those forms may occur on its crystals that possess the same grade of symmetry. Crystallographic Systems. All crystal forms possessing the same grade of symmetry and all forms that may be regarded as derived from these by the suppression of a certain definite propor- tion of their planes in accordance with certain definite laws (par- tial forms) are grouped together as a system. Two . Kinds of Planes of Symmetry. Thus far we have discussed planes of symmetry without distinguishing between the different kinds. As a matter of fact, two kinds must be distinguished principal and secondary planes. Principal planes of symmetry contain two or more equivalent and interchangeable directions directions that may be imagined as interchanged, without affecting the shape of the crystal through which they pass. They are symmetry planes to which two or more other symmetry planes are perpendicular. FIG. 1 9. ^-Crystal Secondary planes of symmetry possess no possessing principal and . secondary planes of interchangeable directions. In the form represented by figure 19, the plane a, a', a, a 7 , is a principal plane of symmetry, while the plane a,c, a, c is a secondary plane. The directions a a, and a' a' may be interchanged by a horizontal revolution of the crystal through an arc of 90 without affecting the shape of the crystal in the least. On the other hand, the directions repre- SYMMETRY 2Q sented by the lines a aandc c are not equivalent directions, because if the crystal were revolved about a' a' through an arc of 90 until c c' takes the direction now held by a a, its long axis would be horizontal, whereas originally it was vertical.* Number and Characterization of Crystal Systems. It . can be proven mathematically that all the crystal forms that are possible may be divided into 32 groups or classes, each charac- ' terized by a special grade of symmetry. To each class a dis- tinctive name is given which indicates its grade of symmetry. It can also be shown that of these groups there are six from which the other 26 may be considered as derived. There are, therefore, six systems into which all crystals may be grouped. Each system includes, therefore, not only the forms possessing a certain grade of symmetry, but also all forms that may be considered as derived from these; i.e., all forms that may be referred to N the same system of coordinates or axes. The groups or systems generally recognized are as follows : System Forms Isometric With 3 principal and 6 secondary planes of symmetry, and partial forms derived from these. Hexagonal With i principal and 6 secondary planes of symmetry, and partial forms derived from these. Tetragonal .- With i principal and 4 secondary planes of symmetry, and partial forms derived from these. Orthorhombic .... With o principal and 3 secondary planes of symmetry, and partial forms derived from these. * It is absolutely necessary that the student should understand the distinction between principal and secondary planes of symmetry before proceeding further. An excellent exercise for this purpose is to separate a number of crystal models, such as are recommended in the footnote, page n, into groups in accordance with the crystal system represented by them. 30 GEOMETRICAL CRYSTALLOGRAPHY Monoclinic With o principal and i secondary plane of symmetry, and partial forms derived from these. Triclinic With no plane of symmetry, but only a center of symmetry, and partial forms derived from these. CHAPTER V. THE ISOMETRIC OR REGULAR SYSTEM. Division of the System. The forms belonging to the iso- metric system like those belonging to all the other systems may be separated into two groups, viz., holohedral and partial forms. The holohedral (oAos, whole, and efyxx, face) forms are those possessing all the planes demanded by the complete sym- metry of the system. A form of this kind is known as a holohedron. The partial forms are made up of the same planes as the corre- sponding holohedral forms, but some of those demanded by the complete symmetry of the system are not present. Partial forms are of a lower grade of symmetry than are the holohedrons. In some cases only half of the whole number of planes found in the holohedron occur on the partial form. This condition is known in some instances as hemihedrism, in others as hemimorphism. In other cases only 1/4 of the planes persist. Such a condition is known as tetartohedrism. HOLOHEDRAL DIVISION. (Hexoctahedral Class.) Complete Symmetry of the System. The holohedral forms of the isometric system possess three principal planes of symmetry at right angles to one another and six secondary planes which bisect the right angles between the principal planes (Fig. 20). In addition to these 9 planes of symmetry the holohedrons of this system possess also 3 principal axes of fourfold symmetry that are perpendicular to the three principal planes of symmetry, 4 axes of threefold symmetry passing through the centers of the octants included between the three principal planes of symmetry, 6 similar axes of twofold or binary symmetry perpendicular to the secondary planes of symmetry, and one center of symmetry. 3 2 GEOMETRICAL CRYSTALLOGRAPHY The presence of the center of symmetry demands the presence on the forms of pairs of parallel planes on opposite sides of the center of symmetry, which must be at the geometrical center of the form. Figure 21 represents in a schematic way the elements of symmetry in the holohedrons of this system. In this and in corresponding figures on later pages the characters of the axes of symmetry are represented by symbols at their terminations as follows: fourfold or tetragonal symmetry; A threefold FIG. 20. The planes of symmetry of the isometric holohedrons. FIG. 21. Elements of symmetry in the isometric holohedrons. or trigonal symmetry; twofold or binary symmetry. Hex- agonal symmetry is represented by . The positions of the prin- cipal planes of symmetry are indicated by heavy lines, and those of secondary planes by lighter lines. Axes. The lines chosen as the axes in this system are the intersections of the three principal planes of symmetry. From the relative positions of these planes with respect to one another the axes must necessarily be three lines perpendicular to each other at a common point. Since, moreover, a secondary plane of symmetry bisects each of the angles between the axes, all these axes must be equivalent; i.e., the axes must be equal in length (or the distances taken as unity on each must be equal) and they must be equally terminated by crystal planes (i.e., by the same number of similar crystal planes). The axes of the isometric system may therefore be defined THE ISOMETRIC, OR REGULAR, SYSTEM 33 as three lines of equal unit lengths perpendicular to each other at a common point (Fig. 13, page 17). Since the axes are imagin- ary lines drawn within a crystal, they extend indefinitely. It must be borne in mind that whenever length is referred to in discussions of crystals relative, and not absolute, length is meant. From the term "equal unit lengths" we must not infer that the unit length in all crystals of the isometric system is the same absolute length, but rather that the ratio between the unities on the axes of all isometric crystals is always one. The Groundform of the System. The groundform of a system has been defined as that form made up of planes cutting the axes at unity distances. In the isometric system the planes of the groundform cut the three axes at the same distance from the center; i.e., the ratio between their in- tercepts is i : i : i. When one plane of this form is present, symmetry demands the presence of seven others, so that the entire form consists of eight planes, which together comprise an octahedron (Fig. 2 2) . Its inter- facial angles are each about 109 28 1/4'. The parameters on the three axes are unity. The symbol of a plane is a : b : c, with the proper signs appended, or, since the axes are equivalent, a : a : a, and the symbol of the form is O. The indices are i i i. The Most General Form in the System. The most general form in the system is that composed of planes cutting the three axes at different distances, no one of which is zero or infinity. Since one of these distances may be considered as unity, the sym- bol of one of its planes is na : b : me. Since, moreover, the different axes are equivalent or since the symmetry of the sys- tem requires that each axis shall be terminated in the same manner, it follows that the presence of the plane na : b : me necessitates the presence of five other planes in the same octant, for without introducing any other than the plus signs understood in the above symbol, this symbol may be changed five times by 3 FIG. 22. Groundform of the isometric system. 34 GEOMETRICAL CRYSTALLOGRAPHY using the same parameters before each of the three axes success- ively, as: na : b : me na : mb : c a : mb : nc a : nb : me ma : nb : c ma : b : nc There are thus in this form six planes in an octant. As there are eight octants in all, the total number of planes in the form must be 48. We can now understand why in this system we may write the symbol of the axes a : a : a instead of a : b : c. Each axis is affected in the same manner by similar planes. There is no character by which one axis is distinguished from the others, and hence we may indicate them all by the same letter. The symbol of the individual planes is na : a : ma, with the proper signs, and that of the form n O m or m O n. Its indices are h k I, in which the three different letters stand for three different values. The Hexoctahedron. The most general form of the system is known as the hexoctahedron (Fig. 23), because it possesses six faces in an octant; or it is a form that may be regarded as an octahedron in which an octahedral face is replaced by six equal faces, each of which is a triangle. Each face cuts one axis at a distance taken as unity, another at n times this distance, and the third at m times this- distance. The interfacial angles are of three FIG. 23. Hexoctahe- kinds (o, c, and d in Fig. 23). Their values dron, mOn, or hkl. yarv W j t j 1 ^ va ] ues o f m an d Hj the figure approaching more and more nearly the habit of the octahedron as m and n approach unity in value. The symbols 203 and 364 both represent hexoctahedra. They signify forms made up of planes cutting the three axes at distances whose ratios are 2:1:3 an d 3:1:4. Their indices are 3 6 2 and 4 12 3. Other Forms Derived from the General Form. In the symbol for the general form na : a : ma, the parameters n and m may be made to vary between o and . If we give n and m all the different values possible, remembering that a plane which does not meet the axis cuts it at o , we obtain the following symbols : THE ISOMETRIC, OR REGULAR, SYSTEM 35 ma : a : ma, when n m; i.e., when both parameters are equal, and neither is unity nor infinity. a : a : ma, or na : a : a, when m or n= i ; i.e., when one of these parameters becomes unity. GOa, when m or n= oo; i.e., when one of these parameters becomes infinity. a : a : a, when m and w=i; i.e., when both parameters become unity. oca : a : GO a, when m and n= o ; i.e., when both parameters be- come infinity. a : a : 000, or co# : a : a, when m orn=i, and the remaining n or w equals infinity. Thus there are only seven possible kinds of forms in the regu- lar system, for a : a : ma and na : a : a are the same kind of form, as are also ooa : a : ma and na : a : l. The Icositetrahedron. The first of these is the icositetrahedron (Fig. 24) or the 24-sided figure, consisting of a polyhedron bounded by 24 similar trapeziums, three in each octant. Each plane cuts one axis at unity and the other two axes at some distance greater than unity and less than infinity. The interfacial angles are two in kind (o and c of Fig. 23). Examples 202 and 3 O 3, or 211 and 311. The Trisoctahedron. The second form, known as the trisoc- GEOMETRICAL CRYSTALLOGRAPHY tahedron (Fig. 25) is bounded by three isosceles triangles in each octant, so distributed that their apices meet at a common point, which is equally distant from the terminations of all the axes. Each plane cuts two axes at unity and one at m and each makes with the contiguous planes two different interfacial angles o and d (of Fig. 23). Examples 2 O and 3 O, or 212 and 313. FIG. 24. Icositetrahe- dron, m O m or ML FIG. 25. Trisoctahe- dron, mO, or hlh. The Tetrahexahedron. The remaining form is also bounded by 24 isosceles triangles (Fig. 26), but they are in groups of four with their apices at a common point which is the termination of an axis. Each plane meets one axis at unity, another at some distance greater than unity, and is parallel to the third. Two different interfacial angles are present. They correspond to the c and d edges of the hexoctahedron. Examples o> O 2, oo 04, 3 O oo ; or 210, 410, and 310. FIG. 26. Tetrahexahe- dron, wOoo orhl. FIG. 27. Dodecahe- dron, oo O or no. Form Composed of 12 Planes. The Dodecahedron possesses 12 planes (Fig. 27), each of which is at the same time in two octants. All are similar rhombs, four of which meet at each termination of the axes, and each cuts two axes at the same dis- tance and is parallel to the third. Their symbols are a : a : ooa. THE ISOMETRIC, OR REGULAR, SYSTEM 37 The interfacial angles are each 120, and since there are no variable parameters possible in the form there can be only one dodecahedron in the system. The symbol of the form is ooQ and its indices no. Form Composed of 8 Planes. The Octahedron, O, has already been discussed. It consists of eight equilateral triangular faces, each in an octant. Its interfacial angles measure 109 28' 16". (See Fig. 28.) Form Composed of 6 Planes. The Cube, ooO^, contains the smallest number of planes possible to holohedrons in the isometric system (Fig. 29). Each of the planes is a square, one of which is perpendicular to each axis at its termination. The interfacial angles are all 90. The indices are 100. FIG. 28. Octahedron, O or in. FIG. 29. Cube, oo Oooor 100. Summary. The total number of holohedral forms possible in the isometric system is seven. Of these, one possesses 48 planes, 3 possess 24 planes each, one possesses 12 planes, one, 8 planes, and one 6 planes. Of these three, the octahedron, the cube, and the dodecahedron, are forms that possess constant interfacial angles, consequently only one of each is possible. The symbols of the other four forms possess variable parameters, hence there may be as many of each of these forms in the system as there are values that may be assumed by their parameters. Their number is limited only by the law of the rationality of the indices. Determination of the Forms. On crystals the symbols that represent the different forms are determined from the interfacial angles made by their planes with each other. By measurement with the goniometer the values of the interfacial angles between selected planes are obtained. From these, by the methods of GEOMETRICAL CRYSTALLOGRAPHY spherical geometry, the value of the intercept on one axis may be calculated in terms of its intercept on another. If the latter be taken as the unity intercept the value determined is the parameter of the plane on the other axis. If in the figure (o>Ow, Fig. 30) the angle C is measured, then by calculation we can determine the relative distances at which the plane t cuts the two axes Oa and Oc. If the distance at which the plane cuts Oa be taken as unity, then the intercept on Oc measured in terms of this unity is the parameter on Oc. If the value of the angle is 126 52', calculation will show that the plane cuts Oc at 3 times the dis- tance at which it cuts Oa. Its symbol is a : oca : 3^ or 3 O a>. Simple Forms and Combi- nations. Each of the figures reproduced above represents a simple form; i.e., a polyhedron all of whose planes are necessi- tated by the presence of a single one. All the planes on each form may be represented by a single symbol. In nature, while crystals occur that are bounded by the planes of a single form, it is more usual to find on them planes belonging to several forms. Such crystals must be represented by as many symbols as there are planes on them belonging to different forms. The total number of planes present is equal to the sum of all the planes belonging to all the forms represented by the symbols. If a crystal contains planes belonging to the forms 203, co O and ooOoo, then the total number of planes on the crystal must be 48+12 +6 or 66. The occurrence of two or more forms on a crystal is known as a combination of forms. The symbol of a combination consists of the symbols of its different forms written in order, with the symbol for the form with the largest faces first. THE ISOMETRIC, OR REGULAR, SYSTEM 39 The forms occurring on the crystal represented in figure 31 are the cube (h) and the dodecahedron (d). Its symbol is ooQoo, oo O. In figure 32 we have combinations of the cube, octahedron, and dodecahedron. On the crystal to the left O predominates, on that to the right oo O oo is the largest form. The symbols for the combinations are O, ooQ oo, ooO and ooO oo, O, oo O. When written in the proper order they indicate the general habit of the crystal which they represent. The first of these two symbols indicates that the crystal which it represents is octahedral in habit, and the second that the corresponding crystal possesses a Cubical habit. FlG 3I . -Combination NOTE. In writing the symbols of com- of cube (ooOco)and do- . . . . . r /- i decahedron (ooO). binations it is necessary first to fix the posi- tion of the axes. This is done by determining the positions of the principal planes of symmetry in the crystal (or model) and noting their lines of intersection. These three lines of intersec- tion are the three axes to which all the planes must be referred. When once fixed they remain the axes for all the planes in the combination. If the choice of the axes is the correct one, their six ends terminate at similar points on the crystal. FIG. 32. Combinations of octahedron, cube and dodecahedron. The next step is to determine the relations of the several faces to these axes. A close inspection of the planes will often suffice to indicate the approximate relative distances at which they cut the axes. If this is not easily seen, a glass plate laid flat 40 GEOMETRICAL CRYSTALLOGRAPHY against a plane will serve to aid in this determination. The distances from the center of the crystal at which the plate cuts the axes, or their prolongations, afford the means for estimating the lengths of the parameters of the plane on which the plate lies. Suppose, for instance, the glass plate has been laid on a plane whose symbol is sought. If the plate meets one axis at a certain distance, a second at some greater distance, and the third at a still greater distance, then, since the shortest of these distances may be made unity, the symbol of the plane is a : ma : na or the form to which it belongs is m O n. Now, suppose the plate be laid on a second plane. It may cut one axis at a certain distance and the other two at similar distances which, however, are different from the distance at which it cuts the first axis. The symbol of the plane is either a : ma : ma , or ma : a : a. The shortest distance is always made unity. Consequently, if the two axes that are cut at the same distance are cut nearer the center of the crystal than is the other axis, the symbol of the plane is ma : a : a, which is a plane of form mO. If, on the other hand, these two axes are cut at a greater distance from the center than the third axis, the plane is a : ma : ma, or a plane of the form m O m. CHAPTER VI. PARTIAL FORMS HEMIHEDRISM AND TETARTOHEDRISM OF THE ISOMETRIC SYSTEM. Independent Occurrence of Partial Crystal Forms. The holohedral forms that have been discussed are those composed of the full number of planes which the complete symmetry of the iso- metric system demands. Sometimes, in consequence of irregular growths, certain planes on a crystal may be developed at the ex- pense of other planes, until finally in extreme cases the latter may be- reduced to mere points, and so disappear as planes. This ir- regular disappearance of some planes of a complete form is acci- dental, and is, consequently, of no crystallographic significance. The condition is known as merohedrism, and the incomplete form is said to be merohedral. In the crystal of quartz (SiO 2 ) represented in figure u the growth was not symmetrical about the axis, and so the crystal is distorted. The plane z at the upper left-hand corner has been crowded by the abnormal development of the planes P and r, until it has nearly disappeared. If the crystal had continued its growth in the same direction, z would probably have disappeared, and the form to which it belongs would have been represented by one less plane than normally belongs to it. Hemimorphism. Occasionally all the planes of half the holohedral form occur at one termination of an axis of symmetry, while from the other end of the same axis they are absent, or are represented by all the planes belonging to half of some other form; i.e., the symmetry axis is polar. It is to be noted that in cases like this all the planes of the given holohedral form that are naturally expected to be present around one termination of an axis are present while none are present at the other end of the same axis. This condition is known as hemimorphism. It is characteristic of a few minerals which always exhibit it and hence must be the direct result of the internal arrangement of their molecules. 42 GEOMETRICAL CRYSTALLOGRAPHY That this is the case is shown by the physical properties of hemimorphic crystals. When these are heated they become pyro- electric, one end becoming charged with positive electricity and the other end with negative electricity. The former is known as the analogue pole, and the latter as the antilogue pole. In writing the symbols of hemimorphic crystals the pole about which the several hemimorphic forms occur must be indicated. (See also pages 216-217.) Figure 33 represents a hemimorphic crystal of calamine (Zn(OH) 2 SiO 3 ), an orthorhombic mineral. The two ends of the c axis are differently terminated, the lower end being terminated by one-half the planes of one form and the upper end by one-half the planes belonging to two other forms. Hemihedrism and Tetartohedrism. In addition to the merohedral and hemi- morphic forms there are other partial forms FIG. 33. Crystal exhib- observed on crystals that are of great im- iting hemimorphism. , . , portance, since their presence characterizes a large number of substances. These forms do not possess the symmetry of the holohedrons belonging to any system, but they are bounded by planes that are the same geometrically as those on certain holohedrons from which they may be regarded as being derived. When the planes that persist in the partial form are one-half of those present on the corresponding holohedron, the new form is called a hemihedron, or it is said to be hemihedral. When only one-quarter of the planes persist, the new form is tetartohedral. Law of Hemihedrism and Tetartohedrism. If the only condition of hemihedrism and tetartohedrism were the occurrence of one-half or one-quarter of the planes of the holohedron, the number of new hemihedrons and tetartohedrons that might exist would be very large. Experience shows, however, that only those planes persist in the partial forms which terminate equivalent ends of the crystallographic axes in a similar manner. In other words, in hemihedrons and tetartohedrons which are not hemimorphic, HEMIHEDRISM OF THE ISOMETRIC SYSTEM 43 equivalent terminations of the crystallo graphic axes are equivalently terminated. That is: All equivalent terminations of the axes in hemihedrons and tetartohedrons (terminations that are separated by planes of symmetry in the holohedrons) must be cut by the same number of similar planes. This limitation of the planes that occur in the new forms reduces their number to compara- tively few. We may imagine any hemihedral or tetartohedral form as being produced by the suppression of a certain half or three- quarters of the planes composing the complete or holohedral form and the extension of the remaining planes until they meet. Suppose, for in- stance, that the white planes of the inside figure (Fig. 34) are suppressed and that the shaded planes are extended to intersec- tion: they will then produce the outside figure, which is the hemihedron corre- sponding to the interior holohedron. In this crystal the terminations lettered A, B, C, D are all equivalent in the holohedral r i i i -i i T p FIG. 34. Crystal exhibit- form, consequently in the hemihedral form ing hemihedrism. Outside they must be cut by the same number f rm contains one-half the J planes of the enclosed form. of equivalent planes. An inspection of the figure will show that each is cut by two planes which are equally inclined to each axis. The axis from E to F is not equivalent to the axes terminating at A, B,C,D, hence its extremities are not terminated in the same way as are A,B,C,D. But there is a plane of symmetry in the holohedral form perpen- dicular to this axis, hence its two terminations are equivalent, and consequently in the hemihedron they should be similarly terminated. As a matter of fact, E and F terminate in three planes each, and the planes that cut at E are inclined to the axis at exactly the same angles as those at which the three planes cutting at F are inclined to it. Combinations of Hemihedral and Tetartohedral Forms. As in the case of holohedral forms, combinations of hemihedral and of tetartohedral forms are frequently met with in nature. 44 GEOMETRICAL CRYSTALLOGRAPHY Since, however, the external forms of crystals are but the expres- sions of definite plans of internal structure, it must necessarily follow that combinations of forms are limited to those of the same grade of symmetry. (See statement of Law of Symmetry, page 28.) Holohedral forms combine with holohedral forms; hemihedral with hemihedral, and tetartohedral with tetartohedral. Further, only those hemihedral forms possessing the same symmetry may combine with one another, and only those tetar- tohedrons with the same symmetry. Some holohedral forms, when subjected to the suppression of one-half or of three-quarters of their planes yield by the extension of the remaining planes new forms that are geometrically identical with the originals; i.e., they are hemihedrons that are identical geometrically with holohedrons. Such apparently holohedral forms may combine with hemihedral or with tetartohedral forms that may be considered as having been derived from other holohedral forms by the same method as was applied to the forms that appear to be holohedrons, and thus there is seemingly an exception to the general statement made above. But this supposed exception is not a real one, since the apparently holo- hedral forms are found to possess all the properties of hemihedral ones, except that of shape. HEMIHEDRISM OF THE ISOMETRIC SYSTEM. Number of New Hemihedral Forms in the Isometric System. The whole number of new hemihedral forms that are directly related to the holohedrons of the isometric system is seven. The other hemihedral forms in this system are geometric- ally identical with holohedrons. Grouping of the Hemihedrons. The seven new hemi- hedrons of the isometric system may be grouped into three classes in accordance with their grade of symmetry. They are all of a lower grade of symmetry than the holohedrons of the system, as would be expected from the fact that they possess fewer faces than those demanded by the condition of holohedrism. Some of them have lost all the principal planes of symmetry that are found in the holohedrons, others have lost all the secondary HEMIHEDRISM OF THE ISOMETRIC SYSTEM 45 planes of symmetry, and another has lost all of its symmetry planes of both kinds. The forms of each of the three groups may be considered as being derived from holohedrons by suppressing a certain half of the holohedral planes and allowing the other half to extend until they intersect. In order, however, that the persisting planes may comply with the condition of hemihedrism, which demands that in the hemihedron equivalent terminations of the crystallographic axes must be equivalently terminated, the derivation of the hemihedrons from holohedrons must take place in one of three ways, each of which gives rise to one of the groups already FIG. 35. FIG. 36. FIG. 37. Figures illustrating the three possible methods of derivation of the hemihedrons in the isometric system. Fig. 35, gyroidal, Fig. 36, pentagonal, and Fig. 37 tetra- hedral. referred to. The only three possible ways by which the desired result may be accomplished are illustrated in the three figures (Figs- 35> 36, 37)- The first figure represents the manner in which hemihedrons without planes of symmetry may be derived; i.e., by the suppres- sion of alternate planes on the hexoctahedron. Figure 36 repre- sents the way in which hemihedrons retaining the principal planes of symmetry may be derived; i.e., by the suppression of pairs of hexoctahedral planes that intersect in the principal planes of symmetry, or those holohedral planes that lie in the alternate sections of the 12 included within the secondary planes of sym- metry occurring in the holohedrons. Figure 37 represents the way in which the third group of hemihedrons may be derived; i.e., by the suppression of all the planes in alternate octants. If we imagine either the white or the shaded planes to disap- 46 GEOMETRICAL CRYSTALLOGRAPHY pear and the others to be extended until they intersect, three new forms will result, each of which will satisfy the conditions of hemihedrism. No other method of suppressing half the planes on this form will result in new forms complying with these con- ditions. Thus three distinct kinds of hemihedrism are possible in this system, and only three. The new forms are known as gyroidal, parallel, or pentagonal, and inclined or tetrahedral. Gyroidal Hemihedrism (Pentagonal Icositetrahedral Class). Although gyroidal hemihedral forms are known on crystals, they are rare and consequently they will not be discussed. One of the FIG. 38. Right pentagonal FIG. 39. Left pentagonal icositetrahedron, r ^J^' icositetrahedron, / m n " two gyroidal hemihedrons derived from mOn is represented in figure 38. This i,s known as the right form and has the symbol r- , or r (hkl). The left form, I , or r (M), is 2 2 represented in figure 39. Pentagonal Hemihedrism (Dyakisdodecahedral Class). The different types of pentagonal hemihedrons are two in number. One is derived from the holohedron mOn and the other from mO oo . Both retain the three principal planes of symmetry, the four trigonal axes of symmetry and the center of symmetry. The three axes of fourfold symmetry become axes of binary symmetry. These elements are indicated in figure 40, which indicates the character of the symmetry that would result if all the shaded planes of mOn or all the white ones should disappear. In other words, it represents the symmetry of all the white planes alone or of all the shaded ones. The heavy HEMIHEDRISM OF THE ISOMETRIC SYSTEM 47 solid lines indicate the positions of the planes of symmetry that would remain, and dotted lines the positions of those that would drop out. The Diploids. By the application of the pentagonal method of selection to the planes of the hexoctahedron, two hemihedral forms are produced that differ according to the positions of the original planes that survive. (Figs. 41 and 42.) Both are known as diploids. They are iden- tical in shape, but differ in their posi- tions with respect to the axes. Either one, by the revolution of 90 about either of its crystallographic axes, may be brought exactly into the position of FIG. 40. Diagram illustrating ,, ,1 -r, ,1,1 ,i i - the distribution of the elements the other. Forms that bear this relation of symmetry in pentagonal to each other are said to be congruent, hemihedrons. The one outlined in heavy lines in figure 41 is designated as the positive dyakisdodecahedron, or diploid, and the other (Fig. 42) as fwO^I [mOn 1 and- -. L 2 J L 2 J Their indices are n(hkl) and nfylk). Each form is bounded by the negative one. Their symbols are FIG. 41. Positive diploid or dyakisdodecahedron, FIG. 42. Negative diploid or dyakisdodecahedron, " twenty-four similar trapeziums meeting in three kinds of inter- facial angles. There are, of course, as many pairs of diploids possible as there are hexoctahedra. Their shapes will necessarily vary with the values of m and n. The Pentagonal Dodecahedrons or the Pyritohedrons. The only other hemihedron of this class may be derived from the 4o GEOMETRICAL CRYSTALLOGRAPHY tetrahexahedron. The extension of alternate planes on the holohedron corresponds exactly to the extension of alternate pairs of hexoctahedral planes that intersect in the planes of symmetry. The new forms derived by the extension of alternate planes of the tetrahexahedron are the pyritohedrons (Figs. 43 and 44). FIG. 43. Positive py- ritohedron, + j^M 00 'J O r ir(hlo). FIG. 44. N e g a t i v e pyritohedron, [-^2] or v(hcl). They are congruent forms bounded by twelve similar pentagons meeting in two different interfacial edges. Their symbols are + ]FwO ooH (Fig. 43) and - (Fig. 44). Their indices are x(hlo) and n(hol). Tetrahedral Hemihedrism (Hextetrahedral Class). The extension of all the planes occurring in alternate octants of holohedral forms in the isometric system and the suppression of those in the other octants will produce new forms in all cases except the tetrahexahedron, the dodecahedron, and the cube. These three holohedrons yield hemihedral forms that are geometrically indis- tinguishable from themselves. Of the FIG 45 Diagram iiiustra- f our new geometrical forms thus pro- ting the distribution of the ele- r . ments of symmetry in tetrahe- duced all have a tetrahedral habit, hence the name of the class. The forms are congruent, each holohedron yielding two hemihedrons, which are distinguished as the positive and negative forms. In the tetrahedral hemihedrons the principal planes of sym- HEMIHEDRISM OF THE ISOMETRIC SYSTEM 49 ,metry, the six axes of binary symmetry, and the center of sym- metry have disappeared. The three axes of fourfold symmetry have become axes of binary symmetry. There remain the six secondary planes of symmetry and the four trigonal axes of sym- FIG. 47. Negative hex- tetrahedron, m H or Fig. 46. Positive hex- tetrahedron, + -^2- or K(hkl). metry. These axes are now, however, polar. The forms con- tain no parallel planes (Fig. 45). The Hextetrahedrons. The hextetrahedrons are 24-sided figures bounded by scalene triangles meeting in 3 kinds of edges. FIG. 48. Positive tristetrahe- dron, + -?* or K (hll). FIG. 49. Negative tristetrahe- dron, or ic(hlfy. The crystal axes terminate in the solid angles at which 4 planes meet. Figure 46 is the positive form, +- , and figure 47, the /~\ negative form, - . Their indices are *(hkl} and K.(hkl). The Trigonal Tristetrahedrons. The trigonal tristetrahedrons are composed of half the planes of the icositetrahedron. Each is 5 GEOMETRICAL CRYSTALLOGRAPHY bounded by 12 similar isosceles triangles intersecting in two kinds of edges. The crystal axes terminate in the centers of the long edges. Figure 48 represents the positive form, +- , and figure 49, the negative one, - are *(hll) and "(hll). mOm Their corresponding indices FIG. 50. Positive del- toid dodecahedron, + or K(hlh) FIG. 51. Negative deltoid dodecahedron, mO. OTK(Mh}. The Tetragonal Tristetrahedrons or Deltoid Dodecahedrons. There are two of these derived from the trisoctahedron as indi- cated in figures 50 and 51. They are bounded by 12 trapeziums intersecting in two kinds of interfacial angles. The crystallo- FIG. 52. Positive tetrahedron + or K(III). FIG. 53. Negative tetrahe- dron, or K(III). graphic axes terminate in the solid angles at which four planes meet. Their symbols are H - (Fig. 50) and - (Fig. 51), 2 2 and their indices are "(hlh) and "(hlh). The Tetrahedrons. These forms are derived from the octa- hedron. There are two of them, each bounded by four equi- HEMFHEDRISM OF THE ISOMETRIC SYSTEM 5 1 lateral triangles intersecting in six similar interfacial edges (Figs. 52 and 53). The crystallographic axes terminate in the centers of these edges. Their symbols are H and - , Their indices are *(iii) and K(III). NOTE. The hemihedrons derived from mOm possess tri- angular faces, while the holohedrons are bounded by trapeziums. On the other hand, the hemihedron derived from mO is composed of trapeziums, while the faces of the holohedron are triangles. TABULAR SUMMARY OF ISOMETRIC HEMIHEDRONS. ( The forms with symbols not written as fractions are indistin- guishable from the holohedrons by their shapes, i.e., they are geo- metrically identical with the corresponding holohedrons.) Hemihedrons Jtioionearons Gyroidal Pentagonal Tetrahedral i mOn mOn ] TwOw] mO i i n r.i. 2 2 J 2 mO oo mO oo mO oo 1 mO oo mOm mOm mOm mOm 2 mO mO mO mO 2 ooQ ooO ooO ooO ooO oo ooO oo ooO oo ooO 00 O o % 2 Combinations of Hemihedrons. Hemihedrons combine with each other exactly as do holohedrons. They may combine with other hemihedrons of the same grade of symmetry, but not with those of different grades of symmetry. The combining 52 GEOMETRICAL CRYSTALLOGRAPHY forms may all be hemihedrons with distinctive forms different from holohedral forms, or they may be in part hemihedrons that are geometrically identical with holohedrons, provided the latter are not such as may yield new hemihedrons with the grade of symmetry of the combining forms. In other words, the hemihe- drons in each vertical column of the above table may be found in combination, but not those in different columns, except when the forms are alike. Thus, O cannot combine with , but oo O 2 and ooQoo may be found in combination with any hemihedron. Further, mOm, mO, and O are never found in combination with tetrahedral hemihedrons, nor mO oo with pentagonal ones. When the +, the , the right, or the left forms alone occur in combi- nation there is no difficulty in distinguishing them from the cor- responding holohedral forms, since by counting the planes possessing the same symbol it may be learned whether they are sufficiently numerous to constitute the holohedral form or only one-half this number. By inspection of the crystal represented in Fig. 54 we detect two kinds of planes, A and B. The symbol of the A planes is ooa : a : ooa; that of the B planes is ma : a : oca. There are six of the former present on the crystal and 12 of the latter. The holohedral form ooO oo possesses six planes; the holohedral form mO possesses 24. Hence the crystal is a combination of oo O oo mO. and - 2 When similar + and , or right and left forms are in com- bination with one another, or when they are both present in combination with other forms, it becomes more difficult to dis- tinguish between them and the holohedral form from which they are derived, for by a combination of the two hemihedrons all the planes of the original holohedron are represented, and it is only by a difference in size of the planes of the two hemihe- drons or by some difference in their appearance that their true nature is recognized. Figure 55 represents the combination of + and - . If HEMIHEDRISM OF THE ISOMETRIC SYSTEM 53 the planes of these two forms were of equal size, there would be no geometrical difference between the combination and the holohedral octahedron. The fact that four of the planes are small and four are large and that the small planes occupy the positions of the planes of one tetrahedron while the large ones occupy the positions of the planes of its congruent form, serves as the criterion by which this combination is distinguished "from the corresponding holohedron. FIG. 54. Combination of ooQoo and +-. FIG. 55. Combination of + and _0. Tetartohedrism of the Isometric System (Tetrahedral Pentagonal Dodecahedral Class). Only four new tetartohedral forms are possible in the isometric system, and these are rare. They are derived from the hexoctahedron by the development of three alternate planes in each alternate octant. They are known as positive and negative, right and left tetrahedral pentagonal , , , , . mOn mOn dodecahedrons. Their symbols are > - and / . CHAPTER VII. THE HEXAGONAL SYSTEM. Systems with One Principal Plane of Symmetry. The hexagonal and the tetragonal systems of crystals are characterized by possessing holohedrons with one principal plane of symmetry and several secondary planes. The principal plane is perpen- dicular to the secondary planes (Figs. 56 and 57), all of which intersect in a common line. The differences in the geometrical forms belonging to the two systems arise from the presence of four secondary planes of symmetry in the tetragonal crystals and six in the hexagonal crystals. In each system the line of the intersection of the secondary planes of symmetry is taken as one of the crystallographic axes, FIG. 56. N FIG. 57. Distribution of planes of symmetry in systems with one principal plane. and the intersections of the principal plane of symmetry with alternate secondary planes give the other axes. In each case the first axis differs from the others which are all equal. The two systems agree in possessing one axis which is not inter- changeable with the others. In studying the crystals the latter is always held in a horizontal position. The analogies existing between- the hexagonal and tetragonal system are so close that a careful study of one makes the study of the second very easy. 54 THE HEXAGONAL SYSTEM 55 Symmetry of the Hexagonal System. The hexagonal system includes all crystallographic forms possessing one principal plane of symmetry and six secondary planes and all the hemi- hedral and tetartohedral forms that may be derived from these. The secondary planes intersect each other in a common line and at an inclination of 30. The principal plane of symmetry is perpendicular to these secondary planes (see Fig. 56). Crystallographic Axes. The lines chosen as the axes of the system are the intersection of the secondary planes with each other and' the intersection of i alternate secondary planes with the principal plane of sym- metry. This selection yields three lines inclined to each _ f other at angles of 60, and all perpendicular to a fourth line. The latter is called the vertical axis and the other three the lateral axes (Fig. 58). rr^, i / .M , i FIG. <$8. The system of axes in the The tWO ends Of the Vertical hexagonal system. axis are separated by a plane of symmetry, hence these two ends must be equivalent. The lateral axes are also separated by planes of symmetry bisecting the angles between them, consequently the lateral axes must all be equivalent. But no plane of symmetry lies between the vertical and the lateral axes. These, therefore, are not equivalent ; consequently, while the unities on the three lateral axes must be equal, the unity on the vertical axis has a different value. Designation of the Axes. As has already been explained, in studying crystals of the hexagonal system the lateral axes are held horizontally. The vertical axis thus becomes upright. The former are designated by the letter a, and the latter by the letter c. The scheme of the axes is a : a : a : c. This symbol indicates that the unities on the three lateral axes are the same and that the unity on the vertical axis is of some other value. The signs given to the axes are indicated in figure 58. A model representing the ratios of the unity lengths of the 50 GEOMETRICAL CRYSTALLOGRAPHY axes would be constructed of three straight wires of equal length intersecting each other at angles of 60, and all perpendicular at their point of intersection to a fourth wire of different length. The Groundform and Axial Ratio. The groundform of the system is composed of planes cutting three of the axes at distances that are relatively the same as the unity distances. Such planes cut two of the lateral axes at precisely the same distance from their point of intersection, and the third axis -(which is the axis c) at a longer or shorter distance from this point. (See Fig. 59.) The third lateral axis is cut at some distance other than unity a distance which for the pres- ent we may represent as x. The symbol of a plane of this kind is a : a : xa : c. FIG. 59. Groundform in When one of these planes is present, sym- hexagonal system. metry demands the presence of n others, which together form a double hexagonal pyramid (Fig. 59). Since the unity length of c is different from the unity length of the a axes, it becomes necessary, before an hexagonal crystal can be studied, to determine the ratio between the two unities in order that a standard may be obtained to which to refer the intercepts of other planes on c. This unity is always recorded in terms of the unity on a. It, therefore, represents a ratio between the lengths at which a plane of the groundform cuts one of the lateral axes and the length at which it cuts the vertical axis. Because it is a ratio between standard lengths on the axes, it is known as the axial ratio, and is written in the form of a ratio as a : c = i : 1.0999. This means that the unity on c is 1.0999 times the length of the unity on a. The value of the axial ratio depends primarily upon the groundform chosen, as the inclination of the planes of the groundform to the a and the c axes determines the ratio between the intercepts on these axes. Let ABC and A'B'C' (Fig. 60) be the planes of two ground- forms cutting the axes OA, OB, and OC at A,B,C, and A',B',C, respectively. The ratio between the lengths on OA and OB and on OC will be determined by the inclination of the planes to the THE HEXAGONAL SYSTEM 57 axes the larger the angle made between the plane and the lateral axes, the larger will be the ratio between the intercepts on these axes and that on c; or the larger will be the axial ratio or, in other words, the larger will be the unity on c. Determination of the Axial Ratio. Every sub- stance that crystallizes in the hexagonal system possesses a different groundform. This groundform is always a double hexagonal pyramid, but the inclination of the faces differs for every different substance. Hence, for every hexagonal mineral the axial ratio must be determined before the sym- bols of the planes occurring on its crystals can be calculated. Practically a form whose planes intercept the vertical axis and two of the lateral axes is assumed as the groundform. The relative distances at which one of its planes cuts the a and the c axes are calculated, after x FIG. 60. measurement of the proper interfacial angles, and this ratio be- comes the axial ratio which is accepted by all crystallographers as representing the arbitrary value which shall be regarded as the ratio between the unities on these axes. GEOMETRICAL CRYSTALLOGRAPHY In figure 61 let A represent the groundform of a crystal the axial ratio of which is to be determined, and DEF, the inter- facial angle between two faces measured at the termination of the line OE, drawn from the point of intersection of the axes perpendicular to the lateral edge. The distance OX = unity on c, and OK unity on a. Let B be a section through the lines OE and OX, and C, a section through the lateral axes. Repre- sent one-half the measured angle by /?. In the triangle OEX,OX=OE tan p (i) In the triangle OEK,OE = OK sin OKE Since OK = i and OKE = 60, and sin. 60 = . 866 OE=.866 Substituting in (i) we have OX or c . 866 tan /? That is, the natural tangent of one-half the measured lateral interfacial angle on the ground-form pyramid multiplied by . 866 is the axial ratio. In the mineral quartz (SiO 2 ) the angle between the planes p and r is 141 47' (Fig. 62). From this value the angle between p and z (below) is easily calculated as 103 34'. One-half of this, or /?, =51 47', and its natural tangent is 1.27. Substituting in the equation above we have i. 27 X .866=1.0998, which is the axial ratio for all crystals of quartz. There are thus as many different axial ratios as there are substances crystallizing in the system. Consequently the axial ratio is a most important distinguishing characteristic of hexagonal substances. For nine important hexagonal minerals it is as follows: Quartz i : 1.0999 Dolomite i : .8322 Nepheline i : .8389 Calcite i : .8543 Cinnabar i : 1.1453 Beryl i : .4988 Apatite i : .7346 Hematite i : 1.3650 Tourmaline i : .4481 The Intercept on the Third Lateral Axis. The ground- form in the hexagonal system is a double pyramid each of whose planes cuts two of the lateral axes at the same distance, assumed as unity, and the vertical axis at some different distance which FIG. 62. Crystal of quartz. THE HEXAGONAL SYSTEM 59 is^also taken as unity. The symbol of its plane as given above (page 56) is a : a : xa : c. As usually written the symbol is a : a : c, or P. The intercept on the third axis is generally omitted from the symbol because its value is determined by the intercepts on the n other lateral axes, in such a way that x is always - , where n = n i the intercept on the first a axis. In the symbol a : a : xa : c, n= i and x=-= <*>. The plane is parallel to the third a and its o complete symbol is a : a : ooa : c. The oo in the symbol is omitted because if the intercepts on two of the a axes are known the intercept on the third axis is also known. In using the indices to designate forms in this system the alternate ends of the axes are considered positive and the interven- ing ends negative as indi- cated in figure 58. The intercepts are written in the order +a l +a 2 , a 3 , c, and the symbol becomes hikl. In all symbols of hexagonal forms the sum of the indices on the three lateral axes 1 7 , , L FlG - 6 3- always = zero; i.e.,. '+*:+ = o. Consequently, if two of the indices are known, the third can easily be deduced. Proof that the Intercept on the Third Lateral Axis is n , when the Intercepts on the Other Two Lateral Axes n i are n and Unity. In figure 63 let AO, BO, and OC represent the lateral axes intersecting at O in the center of a crystal, and let GH be the intersection of a pyramidal face with the plane of these axes. This plane intercepts the three axes at OG, OB, and OH, respectively, and OG, OB, and OH are its parameters on these axes. If OG=n, and OB = unity =i, then OH, the n parameter on the third axis, is - . n i 60 GEOMETRICAL CRYSTALLOGRAPHY Inscribe a circle with O as the center and a radius equal to OB. Then will OA, OB, OC, etc., be the unity distances on these axes. Draw the line B A from the unity distance on B O to the unity distance on AO. In the triangle AOB, the sides OA and OB are equal, . * . the angles OAB and OBA are equal and each is 60. The angle GAB is therefore 120, as is also the angle AOH; . . AB is parallel to OH. Moreover, the triangle is equilateral and AB = OB = i. The triangles GAB and GOH are similar . . GA : GO : : AB : OH, or GA : n : : i : OH. But GA=GO- AO=n i, . * . the equation becomes n i :n : : i : OH. Mul- n tiplying extremes and means gives OH X n i =n, or OH = . n i OH is the intercept on the axis OC, thus the parameter on the third axis is , when the parameters on the other two axes are n i n and i. . Symbols of Forms. In this system and in all of the succeed- ing systems the groundform is represented by P. Unity inter- cepts are not represented. Other intercepts on the c axis are written in front of P, and one of the lateral intercepts, provided these are not unity, is written after the P. In practice the smaller of the two lateral intercepts, when they are not equal, is indicated, the larger one being understood. Thus -a : a : $a : $c is written 3 P -, it being understood that when one lateral intercept is - 4 4' the other is 5. HOLOHEDRAL DIVISION. (Dihexagonal Bipyramidal Class.) Symmetry of the Holohedral Forms. The holohedral forms of the hexagonal system, like the holohedral forms of the isometric system, possess the symmetry which is described as characterizing the system. In addition .to the planes of symmetry described (Fig. 56) the holohedrons of this system possess a sixfold axis of symmetry perpendicular to the principal plane THE HEXAGONAL SYSTEM 6l of symmetry and six axes of binary symmetry at right angles to the sixfold axis, and a center of symmetry (Fig. 64). Figure 65 exhibits the symmetry relations projected on the plane passing through the lateral axes, i.e., upon the principal plane of symmetry. The Most General Form. The most general form of the system is composed of planes which cut the c axis at some dis- tance other than unity and the three lateral axes at different FIG. 64. FIG. 65 Diagram illustrating the distribution of the elements of sym- metry in hexagonal holohedrons. distances. Since one of the latter may be considered as unity, the Naumann symbol of a plane of the most general form is na : a : a : me, where n is greater or less than - , provided n i n i neither is unity. When either n or n n i is unity, or when n n= , the plane will cut two of the lateral axes at the same n i distance, and hence cannot belong to the most general form in the system. \ n The Dihexagonal Bipyramid. Whenw ^- , and neither is n i unity, the planes cut one of the lateral axes at unity, another at n and the third at The presence of one of these planes 62 GEOMETRICAL CRYSTALLOGRAPHY necessitates the presence of 1 1 others on one side of the principal plane of symmetry, and twelve on its opposite side. In all there are 24 planes on the form, one in each of the 24 compartments into which the symmetry planes of the system divide space. The form thus produced is a double pyramid (Fig. 66) bounded by 24 similar scalene triangles. It is known as the di- hexagonal bipyramid. The symbol of one of its planes is na : a : a = me, or if the c axis is n i cut at the distance corresponding to the axial ratio, the symbol is na : a : a : c, and the n i symbol of the form is mPn, or Pn. Its indices FIG. 66. Dihex- are ^- As has alread 7 been stated (page agonal bipyramid, ^Q) ^ it makes no difference whether the shorter mPn or hikl. Qr the i onger o f t j ie two parameters that refer to the a axis is written after the P in the symbols of the forms in this system, as the value of each of these parameters depends upon the value of the other. The symbol 2 P -- and 2 P 3 repre- n sent exactly the same form, for n necessitates , and wee -versa. n i If n = 3, then " = 3 ; if n = -, then ' = r = 3- It is simply n-i 2 2 n-i 1/2 custom that requires the use in the symbol of the smaller of the two parameters that are not unity. The Dihexagonal Series. A series of forms embraces those that may be derived from each other by a change in the value of a single parameter. The dihexagonal series is the series of forms that may be derived from the dihexagonal bipyramid by changing the parameter on the c axis. The symbol of a plane of the most general form is na : a : a : me. By changing the parameter on c we may obtain the n i following symbols: n (i) na : a : a : me. n i THE HEXAGONAL SYSTEM 63 U (2) na : a : a : c. n i n i m na : a : a : c. n-i m n (4) na : a : a : <*>c. (5) na : a : a = oc. The planes represented by these symbols all bear the same relation to the a axes. They are, however, differently inclined to the c axis, and so produce forms that differ from the dihex- agonal bipyramid according to the distance at which their planes cut the vertical axis. The planes represented by the symbols (i), (2) and (3) are di- hexagonal bipyramids. When either (2) or (3) occurs alone it cannot be distinguished from the most general form (i) unless the value of the unity on the c axis is known. When, however, this unity is known, a calculation of the values of the intercepts on c will discriminate between the three forms. For example, one may have three dihexagonal bipyramids of the same substance. Measurement of their interfacial angles may show that a plane on one cuts the c axis at 2 . 1998 times the shortest distance at which it cuts the a axes, which distance is taken as the unity on these axes; that a plane on the second crystal intercepts the c axis at 1.0999 times the unity on a; and that a plane on the third crystal intercepts c at . 54995 times this unity on a. Evidently the three forms to which these planes belong are different and should be represented by different symbols. But there is nothing on the crystals themselves to guide us in the selection of the symbols. If, however, we know that the crys- tallized substance is quartz, the solution of the difficulty is easy. The axial ratio of quartz is i : 1.0999; i- e -> the accepted relation between the unities on a and c is as i : 1.0999. The first of the three forms discussed is composed of planes that cut the c axis at twice this ratio, hence its symbol is 2 Pn. The symbol of the form on the second crystal is Pn, and that on the third crystal, 1/2 Pn. 6 4 GEOMETRICAL CRYSTALLOGRAPHY The Dihexagonal Prism, The fourth symbol, na : a a ni : oo c represents a plane that differs from a plane of mPn in that it is parallel to the c axis. The resulting form is a 1 2-sided prism (Fig. 67), the cross section of which is the same as the cross section of the dihexagonal bipyramid along the principal plane of sym- metry. Its symbol is &Pn and its indices are hiko. Basal Pinacoid. The fifth symbol, na : a : a: oc, stands n i for a plane that cuts the c axis at its point of intersection with the other axes, and the a axes all along their lengths. In other words, it is a plane parallel to the principal plane of symmetry. It is known as the basal pinacoid. Its symbol is oP, and indices, oooi.. The symbol of this pinacoid requires that the plane shall cut one lateral axis at unity and the other two at n and - , respectively, n i and at the same time that it shall pass through the intersection of the axes. A plane that FIG. 67. Dihex- cu t s either of the lateral axes at any distance, agonal prism, ooPw or .... . . . . provided it passes through their intersection with the c axis, must cut them at all dis- tances, hence it is not necessary to specify in the symbol any definite distance at which it cuts these a axes, since all distances are understood. The basal pinacoid is a pair of planes perpendicular to the c axis. Two of them comprise the form, since the existence of one on either side of the plane of symmetry necessitates the existence of one more on its opposite side. In nature trie-planes do not actually pass through the center of the crystals, since this is a physical impossiblity. They are found at the terminations of the c axis, being parallel to the imaginary plane passing through the center of the crystal. (See Fig. 73.) Three Series of Holohedrons in the Hexagonal System. The three forms already noted, viz., the dihexagonal bipyramid, and prism, and the basal plane, constitute the dihexagonal THE HEXAGONAL SYSTEM series. Besides this series there are two others possible in the system. The dihexagonal series results when all of the lateral axes are cut at different distances. When two of the lateral axes are cut at the same distance, two other series arise, according to whether n n in the general symbol na : a : a : me, n = , or either n, or n i n i n = unity. n i When n = i , = oo ; and, conversely, when = i , n = . n i n i n When n = ,n=2: for clearing the equation of fractions we n i have n(n i)=n. Dividing by n this is reduced ton i=i, when n=2. These two changes in the general symbol are the only two that can possibly give rise to new forms from which new series may be derived by changes in the parameters on the c axis. Pyramids and Prisms of the First Order. A plane with the symbol a :a : o>a : me, cuts two of the lateral axes at unity and the c axis at some other distance than the unity on this axis. The third lateral axis is cut at oo. When one plane with this symbol is present, symmetry requires the presence of eleven other similar planes (Fig. 68), resulting in a twelve-sided bipyramid composed of six isosceles tri- angular faces above the principal plane of symmetry, and the same number of similar faces below it. It is known as ~ TO FIG. 68. Hexagonal ra- the hexagonal bipyramid. Its symbol is pyramid of the first order, mP. The indices are ho hi. P or h<*i. From an inspection of the cross section of this pyramid it will be noted that the lateral axes terminate in the solid angles between four contiguous pyramidal faces. Forms that bear this relation to the axes are known as forms of the first order. 66 GEOMETRICAL CRYSTALLOGRAPHY From this pyramid a series of pyramids and a prism are derived by changing the intercept on c, as follows : (I) a : a : a 3 a : mc mP (2) a : a : a >a : c = P i i (3) a : a : a >a;-i-c=- P m w (4) a : a : a a : ooc= a>P (5) a : a '. & ># : oc = oP The symbols 1,2, and 3 represent pyramids of*the first order that differ from each other in the relative lengths of their vertical FIG. 6g. Hexagonal prism of the first order, ooP or lolo. FIG. 70. Hexagonal bipyramid of the second order, mP2, or hbzhl. dimensions. Symbol (2) represents the groundform of the sys- tem. The pyramid mP is more acute than this, and the pyramid P more blunt. m Symbol (4) represents a prism of the first order (Fig. 69). This is composed of six faces with the axes terminating in their interfacial edges. Its symbol is cP or 1010. The plane a : a : oo# : oc does not differ in character from H the plane na : a : a: oc, both, therefore, are represented by the same symbol; i.e., oP. Pyramids and Prisms of the Second Order. The symbol 2a : a : 2a : me, or h h 2h I, designates a plane that cuts two of the lateral axes at twice unity, a third at unity, and the c axis at m. THE HEXAGONAL SYSTEM 6 7 This plane belongs to a bipyramid of 12 faces that is identical in appearance (see Fig. 70) with the hexagonal bipyramid of the first order. The difference between the two pyramids is simply in their relation to the axis. Whereas in the form of the first order the axes terminate in the solid angles made by the conjunction of four faces, in the pyramid of the second order they terminate in the centers of the lateral edges. From the most general symbol of the series four other symbols are derived by changing the parameter on c : (I) 2a : a : 2a : me = ml (2) 2a : a : 2a : c = P 2 i T (3) 2a : a : 2a = _ I ' m m (4) 2a : a : 2a : ooc ', ~~~ QO J (5) 2a : a : 2a : oc = oP Symbols (i), (2), (3), represent three bipyramids of the second order, in which the intercepts on the vertical axis are u~ FIG. 71. Hexagonal prism of the second order, coP2 or 1120. FIG. 72. Diagram illustrating the rela- tions of holohedral bipyramids and prisms to the axes in the hexagonal system. different. Symbol (4) represents an hexagonal prism of the second order in which the axes terminate in the centers of the faces (see Fig. 71), and symbol (5) represents the basal pinacoid. The relations of the pyramids of the three series to the lateral axes are seen in the diagram (Fig. 72), which represents cross sections through the pyramids of the three orders in the plane of the lateral axes. The inner hexagon corresponds to forms of the first order, the outer one to those of the second order, and the 68 GEOMETRICAL CRYSTALLOGRAPHY dodecagon between the two hexagons to the dihexagonal forms. The three solid lines intersecting at the center of the figure are the axes. In practice the distinction between forms of the first and second orders can be made only after the position of the axes has been determined. The selection of the axes is, moreover, merely a matter of convenience in many instances, those lines being chosen as the axes which will yield the simplest symbols for the forms occurring on the crystal under investigation. When the axes are once chosen, however, they remain fixed for all crystals of the substance and all planes must be referred to them. FIG. 73. Hexagonal prism (coP) terminated by basal plane (oP). FIG. 74. Combina- tion of hexagonal prism and bipyramid of same order. FIG. 75. Combina- tion of hexagonal prism and bipyramid of dif- ferent orders. Closed and Open Forms. While each one of the several forms belonging to the isometric system will alone enclose space, this is not true of all the forms of the hexagonal system. The bipyramids may occur alone on a crystal, since each one com- pletely encloses space. The prisms and the basal plane differ from the bipyramids in that neither can completely enclose space and therefore neither can exist alone. They occur either in combination with each other (Fig. 73) or in combination with some other form. Forms that enclose space completely are often spoken of as dosed forms, while those which do not completely enclose space are known as open forms. Crystals bounded by open forms cannot be represented by less than two symbols. Combinations. Combinations of holohedral forms in the hexagonal system are not as common as those of hemihedrons. THE HEXAGONAL SYSTEM 6 9 Where they occur they are easily understood, provided it is remembered that the axes determined for one form are the axes to which all the forms in the combination must be referred. Figure 74 represents a combination of a prism and a pyramid belonging to the same order; figure 75, a prism and a pyramid FIG. 77. Crystal of beryl con- taining ooP (a), P(p), 2P2(r), 2P(w), 3Pf (v), and oP(c). FIG. 76. Combination of 2 hexagonal pyramids of the same order. of different orders; and figure 76, two pyramids of the same order. Figure 77 is more complicated. It represents a beryl (Be 3 Al 2 (SiO 3 ) 6 ) crystal on which are the forms ooP (a), HEMIHEDRAL DIVISION. Hemihedrism in the Hexagonal System. Although the hemihedral divisions of this system are as important as the holohedral division, we must limit our discussion to a few of the simplest of the hemihedral forms. These are extremely impor- tant, since some of the commonest minerals, like calcite (CaCO 3 ), dolomite (MgCaCO 3 ), and apatite (Ca 3 Cl(PO 4 ) 3 ), possess them. Possible Kinds of Hemihedrism. It will be remembered that the lateral axes are not equivalent to the vertical axis in this system. Hence the law of hemihedrism does not require that in the hemihedrons the ends of the vertical axis shall be ter- minated in the same way as the ends of the lateral axes. It demands simply that the new hemihedral forms shall affect the two ends of the vertical axis similarly and the several ends of the lateral axes. 70 GEOMETRICAL CRYSTALLOGRAPHY The number of possible ways by which one-half the planes of the general form in this system may be combined is even greater than in the isometric system. In only four cases, however, does the result comply with the conditions of hemihedrism. These are indicated in the accompanying figures, in which all the white planes are represented as surviving or all the shaded planes. The four classes of hemihedrism derived by these four methods possess different grades of symmetry. The names applied to them are: The Rhombohedral class (Fig. 78) in which the hemihedrons possess all the faces of the holohedrons that lie in alternate dodecants. FIG. 78. FIG. 79. FIG. 80. FIG. 81. Figures illustrating the four possible methods of derivation of the hemihedrons in the hexagonal system. Fig. 78, rhombohedral; 79, pyramidal; 80, trapezohe- dral; 81, trigonal. The Pyramidal class (Fig. 79) in which the alternate pairs of planes meeting in the lateral edges survive. The Trapezohedral class (Fig. 80) in which the planes that survive occupy the alternate compartments of. the 24 into which space is divided by the planes of symmetry. The Trigonal class (Fig. 81) in which all the planes in alternate sextants survive. Only rhombohedral and pyramidal hemihedrons have been found on crystals in complete forms that are geometrically distinct from holohedrons, but hemimorphic forms of the trigonal class are also known. Rhombohedral Hemihedrons (Trigonal Scalenohedral Class). The rhombohedral hemihedrons may be regarded as derived from holohedrons by the suppression of all the planes THE HEXAGONAL SYSTEM 7 1 FIG. 82. Diagram illustrating the distribu- tion of the elements of that lie in alternate dodecants. The new forms thus derived lose the principal plane of symmetry, the three secondary planes passing through the lateral axes, and the three axes of binary symmetry lying between these crystallographic axes. The remaining elements of symmetry, viz., the three secondary planes lying between the lateral axes, the three axes of binary symmetry coinciding with these crystallographic axes, and the center of symmetry survive. The axis of sixfold symmetry becomes an axis of threefold symmetry. This coincides with the vertical crystal axis. (Fig. 82.) In Fig. 82, which is a diagrammatic repre- sentation of these symmetry relations, the heavy broken line indicates the disappear- ance of the principal plane of symmetry. The lighter broken lines indicate the posi- tions of the secondary planes that have disappeared, and the light unbroken lines the position of those retained. Only two new geometrical hemihedrons are possible in this class. The Scalenohedrons. The extension of the planes of the dihexagonal pyramid occupying alternate dodecants yields a form bounded by twelve similar scalene triangles. (Figs. 83 and 84.) These intersect in two kinds of interfacial edges extending from the ends of the vertical axis to the plane of the lateral axes, and a third kind connecting the ends of the lateral axes in a zig-zag line. The lateral axes terminate in the centers of these edges. The forms are known as hexagonal Scaleno- hedrons. Their symbols are . Two congruent scalenohe- 2 drons may be derived from every dihexagonal bipyramid. They are distinguished as the positive (Fig. 83) and the negative (Fig. 84) forms. Their indices are *(hikl) and *(ihkl). The Rhombohedrons. By suppressing every alternate plane composing the pyramid of the first order and extending the remaining planes two new congruent forms are produced, each bounded by six similar rhombs. These possess two kinds of solid 72 GEOMETRICAL CRYSTALLOGRAPHY angles, of which two are polar and four lateral. The polar angles are larger or smaller than the other four depending upon the value of the intercepts on c. When the axial ratio is i : \/i-5 FIG. 83.? o s i - FIG. 84. Nega- tive hexagonal sea- t i v e hexagonal lenohedron, + E scalenohedron, or *(*/). -*f" or *(*/). or i : 1.22474, the polar and lateral angles on the unity rhom- /p\ bohedron ( - ) are equal. The new forms are called the posi- \2 / tive (Fig. 85) and the negative (Fig. 86) rhombohedron FIG. 85. Positive FIG. 86. Negative rhombohedron, + ' rhombohedron. m ^' 2 a or + R, or K(hohl). or R, or K(ohtil"). the latter being the one that turns an upper edge toward the observer. Their symbols are + and , and their in- 2 2 THE HEXAGONAL SYSTEM 73 dices x(hohl) and i<(ohhi). In them the axes terminate in the centers of the six lateral edges. Short Symbols for the Scalenohedron and the Rhombo- hedron. The two rhombohedral hemihedral forms are so frequently met with in nature that many crystallographers use simpler symbols than those given above to represent them. The rhombohedron derived from P is represented by R. That derived from mP is wR, etc. The scalenohedron is represented by R/>, mRp, etc., in which R and wR signify the rhombohedron with the same lateral edges as the scalenohedron (rhombohedron of the middle edges, Fig. 87) and p the distance at which the scalenohedral faces intersect the vertical axis in terms of the corresponding intersection of the planes of the rhombohe- dron of the middle edges. The m and p in this symbol, therefore, do not correspond to m and n in the symbol - . The symbols may be transformed FIG. 87. Hexag- onal scalenohedron, enclosing rhombohe- dron of the middle edges. from one system into the other with the aid of the following equations:* mPn m(2 n) ' n i P 1/2 2 n 2 n' mpP 2 p P + Combinations. The combinations of the pyramids and prisms of the first and second orders with the rhombohedrons are at first confusing, but a little consideration of the position of the axes will nearly always serve to make them clear. Figure 88 represents a combination of R with the prism of * For proof of the correctness of these equations see Groth: Physikalische Krystallographie, 1885, p 342 and p. 348. 74 GEOMETRICAL CRYSTALLOGRAPHY the first order; figure 89, a combination of +R with the prism of the second order; and figure 90, that of a +R with a R. If the FIG. 88. Combi- FIG. 89. Com- FIG. 90. Combination of nation of negative bination of posi- positive and negative rhom- rhombohedron( R) tive rhombohe- bohedrons. with prism of the dron ( + R) with first order ( ocP). prism of the second order positions of the axes are fixed with reference to the rhombo- hedrons in these crystals, the nature of the prisms is easily recognized. The crystals represented in figures 91 and 92 are FIG. 92. Crystal of cal- ate containing R(/>),|R(s), 4 R(m),R 3 (r)R 5 (;y),RH?'), and R 3 0- FIG. 91. Hematite (Fe 2 O 3 ) crystal containing R(R), R(r) and-iR(n). more complicated. The first contains planes of the forms R(r), and - --R (n). Figure 92 represents a crystal of calcite (CaCO 3 ), with the forms R (/>), 5 R (Y), 4R (m) and the scaleno- hedrons R 3 (r), R 5 (y), R* (v), and -R 3 (/). 4 THE HEXAGONAL SYSTEM 75 Pyramidal Hemihedrons (Hexagonal Bipyramidal Class). The pyramidal hemihedrons may be derived from the holohedrons by extending the alternate pairs of planes meeting in the lateral edges. They are characterized by the presence of a principal plane of symmetry, an axis of sixfold symmetry, and a center of symmetry. The secondary planes of symmetry and the six axes of binary symmetry of the holohedrons are lost (Fig. 93). The only holohedrons from which new pyramidal hemihedrons may be derived are the dihexagonal bipyramids and prisms. The Pyramids of the Third Order. The dihexagonal bipyramid gives rise to two bipyramids of the third order which FIG. 93. Diagram illustrating the distribu- tion of the elements of symmetry in pyramidal hemihedrons. FIG. 94. Positive hex- agonal bipyramid of the third orderj + [^\ or ir(hikl}. FIG. 95. Negative hex- agonal bipyramid of the third order, - f 2 ^] or TT (hkit). are like the pyramids of the first and second orders in possessing 12 similar triangular faces and a hexagonal cross section (Figs. 68 and 70). They differ from these bipyramids, however, in their relation to the lateral axes. In the bipyramid of the third order the axes terminate in the lateral edges somewhere between the centers of the edges and the solid angles. The two forms are congruent and their symbols are + \mPn\ /T \Pn\ \rnPn] + (Fig. 94), - Y > ~^~ r g ' 95 ^ ' L . ^ " J L_1L J indices are 7t(hikl) and n(hkil). The Prisms of the Third Order. In the same way the dihex- agonal prisms yield prisms of the third order (Fig. 96), which, GEOMETRICAL .CRYSTALLOGRAPHY when observed alone, are not distinguishable by sight from the prisms of the first order. When in combination with other prisms, however, they are easily recognizable from the fact that the lateral axes terminate neither in the interfacial edges between contiguous planes nor in the centers of these planes. Their symbols are and their indices x(hiko) and x(hkio). The Relations of the Pyramids and Prisms of the Three Orders. The pyramids of the three orders are impossible to distinguish by the eye alone when either occurs singly. If the axial ratio is known, a measurement of the lateral interfacial angle and a calculation from this of the ratio between the distances at which the pyram- idal planes cut the a and c axes will deter- mine the question. If this ratio is the same as the axial ratio, or a multiple of it, FIG. 96-Hexagonal the p yram id is of the second order (c = prism of the third order, r * + ["-?"], orir(hiko). tan /?, where OE=i. Compare p. 58.) If the axial ratio is different from the de- termined ratio, the pyramid is of the first or third order. If it is .866 that of the determined ratio, or some multiple of .866, the pyramid is of the first order (compare p. 58). If the deter- mined ratio is not the same as the axial ratio or some multiple of it, and is not .86, i . 7, or 2 . 5 of it, the pyramid is of the third order. The three orders of prisms are impossible to distinguish either by the eye or by measurement when they occur alone, even if the axial ratio is known. When in combination there is little difficulty in discriminating between the different orders. If the order of any form in the combination is known, it is only necessary to locate the lateral axes to determine the orders of all the prisms and pyramids in the combination. The relation of the orders to the axes is indicated in the subjoined figure (Fig. 97) in which the heaviest line indicates the position of the prism of the third order. The Trapezohedral Hemihedrons (Hexagonal Trapezohe- dral Class). In the hemihedrons of this class all the planes of THE HEXAGONAL SYSTEM 77 symmetry and the center of symmetry have disappeared. The forms therefore possess no pairs of parallel planes. All the axes of symmetry remain (Fig. 98). -IB* / \ / T FIG. 97. Diagram illus- trating the relations of the prisms and,pyramids of the three orders to the hexagonal axes. ** B^ji * FIG. 98. Diagram illus- trating the distribution of the elements of symmetry in h'exagonal trapezohedral hemihedrons. FIG. 99. Right hex- agonal trapezohedron r Eor r(hikl}. FIG. 100. Left hexagonal trapezohedron, l mPn f or r(kihl}. Two new forms are possible. These are known as hexag- onal trapezohedrons. They are enantiomorphous (see page 83), mPn and are represented by the Naumann symbols r - , and ntPn The r and / signify right (Fig. 99) and left (Fig. 100). Since no crystals bearing forms of this class have been ob- served, it is needless to discuss them in this book. GEOMETRICAL CRYSTALLOGRAPHY The Trigonal Hemihedrons (Ditrigonal Bipyramidal Class). -These forms have never been observed except on hemimorphic crystals. The theoretical forms possess one principal and three secondary planes of symmetry, an axis of trigonal symmetry, and three binary axes of symmetry. Three of the secondary planes of symmetry and three of the axes of binary symmetry belonging to the holo- hedrons have disappeared, as has also the center of symmetry. The axis of sixfold symmetry has become an axis of threefold symmetry (Fig. 101). As ob- served in connection with hemimorphism, the principal plane of symmetry and the three axes of binary symmetry are lost and the axis of trigonal symmetry becomes polar. The forms of this class are characterized by triangular or triangle-like cross sections. The Trigonal Prisms and Bipyramids. These hemihe- drons may be derived from the prisms and pyramids of the first order by the suppression of the planes in alternate sextants. The FIG. 101. Diagram illus- trating the distribution of the elements of symmetry in trigonal hemihedrons. FIG. 102. Positive trigonal OOP FIG. 103. Negative trigonal oo P ,7 prism, -^-, or ohho. results are two congruent prisms and two congruent bipyramids with cross sections that are equilateral triangles. Two lateral axes terminate in each of the planes of the prism and in each lateral edge of the pyramid in such a manner that the distance between the polar -edges is divided into three equal spaces. THE HEXAGONAL SYSTEM 79 The new forms are known as the trigonal prisms and bipyra- ooP mids. There are two of each with the symbols H - (Fig. 102) 2 ooP mP mP and - (Fig. 103), and+ - - (Fig. 104) and - (Fig. 105). 222 Their corresponding indices are hoho, ohho and kohl and ohhl. FIG. 104. Positive trigonal bipyramid + , or hohl. FIG. 105. Negative trigonal bipyramid, -^-, or ohhl. The Ditrigonal Prisms and Bipyramids. Similarly the di- hexagonal prisms and bipyramids yield four new forms, two con- gruent ditrigonal prisms and two congruent ditrigonal bipyramids. These possess two kinds of polar edges and their cross sections FIG. 1 06. Diagram illustrating relation of planes in ditrigona pyramid and prisms to the axes in the hexagonal system. are six-sided with the sides arranged in pairs so as to produce a triangle-like outline. The crystallographic axes in the prisms terminate at points between the centers of the faces and the blunt vertical edges. In the bipyramids they terminate at the corre- sponding points on the lateral edges (Fig. 106). 8o GEOMETRICAL CRYSTALLOGRAPHY The symbols of the prism are - (Figs. 107 and 108) and kOj ihko, and of the bipyramids - - (Figs. 109 and no) and> 2 hikl, ihkl. FIG. 107. Positive ditrigonal FIG. 108. Negative ditrig- prism, + ^- n , or hik. Pn onal prism, - , or ihko. Combinations with Hemimorphism (Ditrigonal Pyramidal Class). As has already been stated, the forms of this class are known only in combination with hemimorphism. Consequently only the upper or lower half of each form is present on any one crystal. In the case of the prisms, however, there is no geometrical FIG. 109. Positive ditrigonal bipyramid, + ^- w , or hikl. FIG. no. Negative ditrigo- nal bipyramid, P , or ihkl. difference noticeable between the complete hemihedral forms and those that are hemimorphic, since each plane belongs at the same time to both upper and lower dodecants. In the case of the bipyramids, on the other hand, half of the planes exist either at the upper pole or at the lower one. Consequently there are THE HEXAGONAL SYSTEM 8l four hemimorphic forms of the bipyramid possible. Because the planes on the hemimorphic pyramids are only 1/4 the total number of planes on the corresponding holohedron their mP niP symbols become + -- u, (hohl), -\ ----- / (ohhl), 4 4 -jjjT) wT* u (ohhl) and - / (hohl), in which the 4 4 letters u and / after the Naumann symbols indicate upper and lower. The trigonal hemihedrons are easily recog- nized in combinations by their cross sections. The best illustration of this type of hemi- hedrism is shown by the mineral tourmaline combinations of trig- , ... J N ~ onal hemihedrons. (a complex boro-silicate) ; figure in repre- sents the combination of (n), P P 2P OP -u(P), -f-/(P'), -u(o'), / (c), 4442 / and - P l. TABULAR SUMMARY OF HEXAGONAL HEMIHEDRONS. (See explanation under list of isometric hemihedrons, page 51.) Hemihedrons Holo- hedrons mPn oo Pw 00 P2 mP ooP oP Rhombohedral wPw or mRp oo Pn mP2 mP , orR 2 p TooPw 2 OP mP ooP oP Pyramidal Trapezohedral H em?moL mPn 2 mP ooP oP mPn -- 4 oo Pn mP -u.l. 4 ooP 2 OP 82 GEOMETRICAL CRYSTALLOGRAPHY Combinations of Hemihedrons. The statements made with reference to the combinations of hemihedrons in the iso- metric system (pp. 51-52) apply equally well to those of the hexa- gonal and the succeeding systems. Only those hemihedrons may combine that have the same grade of symmetry; i.e., of the forms indicated above, only those may be found in combination that are represented by the symbols in the same vertical columns. TETARTOHEDRAL DIVISION. Tetartohedrism of the Hexagonal System. The tetar- tohedral forms of this system, like the hemihedral forms, are of great theoretical importance because the most widely spread of all minerals, quartz (SiO 2 ), often exhibits well-characterized tetartohedral planes. Moreover, they are of considerable prac- tical interest because of the fact that there is a close relation existing between certain important physical properties of tetar- tohedral crystals and the planes occurring on them. There are three classes of tetartohedrons in this system distinguished from one another by symmetry. They are known as Trigonal, Trapezohedral, and Rhombohedral tetartohedrons because their characteristic forms are trigonal bipyramids and prisms, trapezohedrons and rhombohedrons. They naturally possess a different grade of symmetry from the hemihedral forms of the same names, and therefore are designated as of different orders in the case of the bipyramids, prisms, and rhombohedrons. The tetartohedral trapezohedrons have^three-sided polar angles, and therefore are distinguished from the corresponding hemi- hedral forms, which have six-sided polar angles, by prefixing the adjective trigonal to the name of the form. Only the trapezo- hedral tetartohedrons will be discussed. Trapezohedral Tetartohedrons (Trigonal Trapezohedral Class). Only ten forms of this class are different geometrically from holohedrons and hemihedrons. Of these four are trigonal trapezohedrons, two are ditrigonal prisms, two are trigonal prisms, and two are trigonal pyramids. The only elements of symmetry in them are an axis of threefold symmetry coinciding with the vertical crystallographic axis and three polar axes THE HEXAGONAL SYSTEM of binary symmetry which coincide with the lateral crystallo- graphic axes. The loss of the center of symmetry means the absence of pairs of parallel planes. (Fig. 112.) The forms may be regarded as being derived from the holo- hedrons by applying to them at the same time the conditions of rhombohedral and trapezohedral hemihedrism (Fig. 113); or they may be thought of as being derived from the rhombohedral hemihedrons by applying to them the condition of trapezohedral he-mi hedrism, or vice versa. FIG. ii2. Diagram illus- trating distribution of sym- metry elements in trapezohe- dral tetartohedrons. FIG. 113 Application of rhombohedral and trapezo- hedral hemihedrism to the dihexagonal bipyramid. Trigonal Trapezohedrons. The trigonal trapezohedrons con- tain six trapezoid faces that correspond to six faces of the dihex- FIG. 114 Positive right trigonal trape- zohedron, + r *^ 4 ' or hikl. FIG. 115. Positive left trigonal trapezo- hedron, + 1**L } or MM. agonal bipyramid. They meet in six equal polar edges and in zig-zag lateral edges made up of long and short lines. Three 8 4 GEOMETRICAL CRYSTALLOGRAPHY lateral axes with the same sign terminate in the centers of the long lateral edges, and the other three in the centers of the short edges. Since each form contains but one-fourth the planes of the holohedron, there are in all four forms derivable from each holohedron (see Fig. 113). These are designated as positive ( +), negative ( ), right (r), and left (/). The + and forms are congruent and the r and / forms are enantiomorphous; i.e., they are symmetrical with respect to one another and neither can be so revolved that its faces shall be parallel to the faces of the other. J77 T"^77 / YM "P 1 !/ The Naumann symbols are +r (Fig. 114), r :L -, 4 4 -f7 (Fig. 115), and I . The corresponding indices are 4 4 ihkl, kihl and khil. FIG. 116. Diagram illustrating relation of planes to axes in hemihedral (A) and tetartohedral (B), ditrigonal prisms and pyramids. \ FIG. 117. Right ditrigonal prism of the second order, * or hiko. 2 FIG. 118. Left ditrigonal prism of the second order,/ ^ "> khio. The Ditrigonal Prisms of the Second Order. The dihexagonal prisms yield two ditrigonal prisms with the symmetry of trapezohe- dral tetartohedrism. These are .geometrically similar to the hemi- THE HEXAGONAL SYSTEM hedral ditrigonal prisms. They differ from them, however, in the relation of their planes to the axes (see Fig. 116). Because of this difference the forms of this class are known as of the second order, and their symbols are written r (Fig. 117) and P 2 /._ (pjg > n8), or hiko and kiho. FIG. 119. Right trigonal bi- pyramid of the second order, r 2 } or hh2hl. FIG. 120. Left trigonal bipyra- mid of the second order, /^L2. ) O r 2hhhl. FIG. 121. Right trigonal prism of the second order, r *P* , or hhzho. FIG. 122. Left trigonal prism of the second order, / - -, or 2hhho. Trigonal Bipyramids and Prisms of the Second Order. The hexagonal bipyramids and prisms of the second order yield trigonal bipyramids and prisms of the second order which differ from the corresponding hemihedral trigonal forms in their positions with respect to the axes the relation of their faces to the axes being the same as in the hexagonal pyramid of the second order; viz., in the prism each face is perpendicular to one 86 GEOMETRICAL CRYSTALLOGRAPHY axis, and in the pyramids each lateral edge is at right angles to an axis (see Figs. 120 and 122). When in combination with other forms the tetartohedral forms are easily recognized by their positions. To distinguish them from other similar geometrical forms they are represented by oo P2 , and r - and / 2 (Figs, no, 120, 121, and 122). 22 Combinations. The mineral exhibiting these forms in greatest perfection is quartz (SiO 2 ). Two crystals of this sub- stance are shown in figures 123 and 124. Crystals showing right forms are referred to as right crystals, and those showing left forms are known as left crystals. The former possess the power of rotating the plane of polarized light to the right, and the latter the power of turning it to the left. Since this effect of quartz FIG. 123. FIG. 124. Crystals of right-handed (Fig. 123) and left-handed (Fig. 124) quartz crystals containing ooP(a), +\(r), j-(z), + ^ and + r ^/-* (s and x in Fig. 123), and _ ?^_ 2 and + /-^6/s ( S and x in Fig I24 ) upon polarized light is made use of in the manufacture of certain optical instruments, the ability to recognize right and left forms is extremely valuable. The forms exhibited by the two crystals P P 2?2 2?2 figured are: 2.4142, and in the more acute ones when w<2.4i42. It is crystallographically impossible that all the solid lateral angles should be equal and all the interfacial polar edges similar, THE TETRAGONAL SYSTEM 91 since in this case the value of the parameter n in the general formula a : na : ma would be the tan of 67 30' which is 2 . 4142 + , and this is irrational. For let the figure 132 represent a cross section of the bipyramid through the lateral axes (OA and OD) and let BC and B'C' be the traces of the two planes a : na : me and na : a : me in the same octant. In the quadrangle OBXB' the angles = 360. The angles B +X+B' = 36o the angle O which is 90. Consequently the sum of B, X and B' = 27o (i). If we assume the interfacial angles to be equal, then X=B + B', in which B and B' each equal half of the interfacial angle at FIG. 132. the ends of the axes, measured in the plane of the axes, and X the total value of the interfacial angle between these two. Sub- stituting the value of X in (i), we have 26+26'= 270. But B and B' are equal, consequently 46 270, or 6 = 67 30'. The intercept of the plane represented by BC is unity (OB) on the axis OA. OC (or the intercept on the axis OD) = OB tan 67 30', or since OB = i, OC = nat. tan of 67 30' which is 2.4142 +. The symbol of the ditetragonal bipyramid is mPn; its indices are hkl. The Ditetragonal Series. A series of forms may be derived from the ditetragonal bipyramid in the same way as the dihex- GEOMETRICAL CRYSTALLOGRAPHY agonal series was derived from the dihexagonal bipyramid, by giving different values to the parameter on c. Thus : (i) a : na : me (3) a : na : c (2) a : na : c m (4) a : na : (5) a : na : oc The first three symbols refer to planes of ditetragonal bi- pyramids that differ from each other merely in the distances at which their planes intercept the c axis. The fourth symbol represents a plane of the ditetragonal prism oePw (Fig. 133), and the fifth, the basal plane oP. Series of the First and Second Orders. In the symbol a : na : we, n may be given different values, which will affect the character of the form produced. If n be given any value FIG. 133. Ditetragonal prism, oo Pn or hko. FIG. 134. Tetragonal prism of the first order, ooP or no. other than unity or infinity the symbol, becomes a : n'a : me, which represents 'a plane of the ditetragonal bipyramid mPn f . When n = oo , we have a : oo a : me, or mP oo , and when n=i, we have a : a : me or P. The first is the symbol of a bipyramid of the second order, and the second that of the first order. Tetragonal Bipyramids and Prisms of the First Order. The symbol a : a : me represents a plane cutting the two lateral axes at the same distance, which may be taken as unity, and the c axis at some distance other than the unity distance on this axis. The form mP, made up of eight of the planes, which are isosceles THE TETRAGONAL SYSTEM 93 triangles, is a bipyramid of the first order (Fig. 129) in which the lateral axes terminate at the solid angles formed by two planes above and two below the principal plane of symmetry. Its indices are hhl. By changing the parameter on c to unity, FIG. 155. Tetragonal bipyramid of the second order, wPoo or ho. FIG. 136. Tetragonal prism of the second order, ooPoo or 100. infinity, and zero, the symbols become P, oo P, and oP. The first is the groundform; the second, the prism of the first order (Fig. 134), and the third, the basal plane. The indices of the prism of the first order are no and of the basal plane ooi. Tetragonal Bipyramids and Prisms of the Second Order. When n in the symbol a : na : me be- comes oo, the symbol becomes a : oca : me. This is a plane of a bipyramid which differs from the bipyramid of the first order only in its position with respect to the axes. These terminate in the centers of the lateral edges, hence, from its analogy with the hexagonal bipyramid of the second order, the form is known as the tetragonal bipyra- mid of the second order. Its symbol is wPoo (Fig. 135). Its indices are hoi. From this form the prism of the second order is derived in the same way that the prism of the first order is derived from the corresponding pyramid. Symbol, Poo. Indices, 100. (Fig. 136.) FIG. 137. Diagram illustrating relations of planes of holohedral pyramids and prisms to axes in tetragonal sys- tem. 94 GEOMETRICAL CRYSTALLOGRAPHY Relations between Pyramids and Prisms of the Various Orders. The relations existing between the various pyramids and prisms of the tetragonal system are shown in the diagram, figure 137, which is a cross section along the principal plane of symmetry. The inner square gives the position of the bipyramid of the first order with respect to the lateral axes, and the outer FIG. 138. Combina- tion of bipyramid and prism of same order. FIG. 139. Combination of bipyramid and prism of different orders. square that of the bipyramid of the second order. The octagon between these squares gives the position of the planes of the ditetragonal bipyramid. Combinations. The combinations of tetragonal forms are similar in character to those of hexagonal forms. When the axial ratio of a mineral is not known, either of its simple tetragonal FIG, 140. FIG. 141. Combinations of bipyramids of different orders. The pyramid of the second order has the shorter intercept on c in Fig. 141 and the longer intercept on this axis in Fig. 142. prisms or bipyramids may be regarded as the form of the first order, and in this way the position of the axes is fixed. Other simple bipyramids and prisms belong to the series of the first order or to that of the second order according to their relations THE TETRAGONAL SYSTEM 95 with these axes. Ditetragonal bipyramids and prisms are always recognizable by the number of their planes. Figure 138 is a combination of a bipyramid and prism of the same order, and figure 139 that of a bipyramid and a prism of different orders. Figures 140 and 141 are combinations of pyramids of the two orders. In the former the ~ , FIG. 142. Crystal of an- pyramid of the second order has a atase containing 1/3 P (>, , ,, ,. r ,, i/7 P(v), ooP(w), ooPoo(a) shorter parameter on c than that of the an ^ p ,(). first order, and in figure 141 it has a longer parameter on this axis. Figure 142 is a crystal of anatase (TiO 2 ) with 1/3 P(z), 1/7 P(V), ooP(w), o>Poo(a), and Poo (e). HEMIHEDRAL DIVISION. Hemihedrism in the Tetragonal System. Although the hemihedral division of this system is by no means as important as the hemihedral division of the hexagonal system, it nevertheless deserves consideration since several well-known minerals exhibit forms belonging to it. Possible Kinds of Hemihedrism. In the tetragonal system there are three classes of hemihedrons distinguished by differences in symmetry. They may be regarded as derived from the holohedrons by the three methods indicated in the subjoined figures. The first method (Fig. 143) is known as sphenoidal hemi- hedrism; the second, pyramidal hemihedrism (Fig. 144), and the third, trapezohedral hemihedrism (Fig. 145). Forms derived by the third method have not been observed on minerals. Those derived by the second method are found on several rare minerals. The sphenoidal forms are seen on the common mineral chalcopyrite (CuFeS 2 ). The Sphenoidal Hemihedrons (Tetragonal Scalenohedral Class). This class of hemihedrons may be considered as being derived from the holohedrons by the suppression of the planes in alternate octants and the extension of all others. This is 9 6 GEOMETRICAL CRYSTALLOGRAPHY analogous to tetrahedral hemihedrism in the isometric system, and rhombohedral hemihedrism in the hexagonal system. In this class of hemihedrons the principal plane of symmetry and the two alternate secondary planes passing through the FIG. 144. FIG. 145. FIG. 143- Figures illustrating the possible methods of derivation of the hemihedrons in the tetragonal system. Fig. 143, sphenoidal; Fig. 144, pyramidal; Fig. 145, trap- ezohedral. lateral axes disappear, as do also two of the axes of binary sym- metry and the center of symmetry. Moreover, the fourfold axis is changed to a binary axis. The remaining elements of symmetry are three binary axes of symmetry perpendicular to one another and coinciding with the crystallographic axes, and the two secondary planes of symmetry passing between the lateral crystal axes (Fig. 146). Four new geometrical forms belong to this class, two derived from the ditetragonal bipyramid and the other two from the bipyramid of the first order. Tetragonal Scalenohedrons. The most general form of the system, mPn, gives rise to two congruent scalenohedrons, each composed of 8 similar scalene triangles meeting in three kinds of interfacial angles and each possessing two kinds of solid angles. The two planes passing through the edges in which the vertical axis terminates are perpendicular to FIG. 146. Diagram illus- trating distribution of symmetry elements in sphenoidal hemi- hedrons. THE TETRAGONAL SYSTEM 97 each other. The lateral axes terminate in the centers of the edges which are not in these planes. mPn mPn The symbols of the forms are + (Fig. 147) and - r (Fig. 148), or n(hkl) and K (hkl). FIG. 147. Positive tetragonal scalenohedron, + *"?- o r K(hkl). FIG. 148. Negative tetrag- onal scalenohedron, n or K(hkl). FIG. 149. Positive tetragonal sphenoid, + -or K (hhl). FIG. 150. Negative tetragonal sphenoid, FIG. 151. Crystal of urea with oP(c), ooP(w) and + -?- (o). Tetragonal Sphenoids. The tetragonal sphenoids contain half the planes of the bipyramid of the first order. They differ in appearance from the isometric tetrahedron in that they are not of equal dimensions along the three axes. The forms consist 7 98 GEOMETRICAL CRYSTALLOGRAPHY of four similar isoscles triangles meeting in two kinds of inter- facial edges. Their symbols are + (Fig. 149) and- (Fig. 150), or 2 2 K(hhl) and K(hhl). Figure 151 shows the combination of oP (c), P ooP(w), and +- (o\. 2 The Pyramidal Hemihedrons (Tetragonal Bipyramidal Class). The hemihedrons of this class may be derived from the holohedrons by the extension of all the planes lying in the alternate sections into which the secondary planes of sym- metry divide them. These forms retain the principal plane of symmetry, the axis of fourfold sym- metry, and the center of symmetry. The FiG.i 5 2.-^iagramillu S - secondary planes and binary axes of trating distribution of sym- symmetry found in the holohedrons dis- metrv elements in pyramidal ,. N hemihedrons. appear (Fig. 152). There are four new forms in this class, of which two are derived from the ditetragonal bipyramid and two from the corresponding prism. The forms are congruent. They correspond to the pyramidal hemihedrons of the hexagonal system and the parallel hemihedrons of the isometric system. Tetragonal Bipyramids and Prisms of the Third Order. These are forms that resemble geometrically the bipyramids and prisms of the first and second orders. They differ from them, however, in the relation of their planes to the crystal axes. The pyramids of the third order are derived from the di- tetragonal bipyramids, hence their Naumann symbols are rwPw~l [mPnl + I and - 1 the brackets indicating that they belong L 2 J L 2 J to the pyramidal class. The prisms of the third order are derived from the ditetragonal prisms, and are represented by the l symbols + and - - . Their corresponding indices 2 2 J THE TETRAGONAL SYSTEM 99 are x (hkl) and K (hkl), and n (hko) and K (hko). The relations of the planes of the positive forms to those of the pyramids and prisms of the first, second, and ditetragonal orders are shown in figure 153. The heavy lines indicate the positions of the planes belonging to the forms of the third order. FIG. 153. Diagram illus- trating relations of planes of pyramids and prisms of various orders to the axes in the tetrag- onal system. FIG. 154. Crystal of stol- zite with P(o) and FIG. 155. Crystal o f scheelite with Poo(<0,P(>),+ [^] (*)and[3|3] (j ). Combinations. In combination the pyramids and prisms of the third order are distinguished from those of the first and second orders by their positions on the crystal. Figure 154 represents a crystal of stolzite (PbWO 4 ) bounded by P(0) and 100 GEOMETRICAL CRYSTALLOGRAPHY + -^ (#), and n g ure i55> one of scheelite (CaWOJ on which occur P oo (), P (o), + j ^ I (h) and - [ ^1 (s). YIG. 156. Diagram illustrating distribution of symmetry elements in trapezohedral hemihedrons. The Trapezohedral Hemihedrons (Tetragonal Trapezohedral Class). By the extension of the alternate planes on the dihex- agonal bipyramid two new forms are derived, and these are FIG. 157. Right tetrag- onal trapezohedron, r ^ n or r(hkl). FIG. 158. Left tetrag- onal trapezohedron, l m ^-, or r(hkl}. enantiomorphous. They retain the axis of fourfold symmetry and the four axes of binary symmetry, but have lost all planes of symmetry and the center of symmetry (Fig. 156). The forms are known as the right and left tetragonal trapezohedrons and THE TETRAGONAL SYSTEM IOI are given the symbols r (Fig. 157), / - (Fig. 158) or T (hkl) and T (hkl) . They are characterized by having four equal polar edges and a zig-zag lateral edge composed of long and short courses. The lateral axes terminate in the centers of the larger zig-zag edges. Because they have not yet been found on crystals they are not discussed in detail. TABULAR LIST OF HEMIHEDRONS IN THE TETRAGONAL SYSTEM. (See explanation under list of isometric hemihedrons, p. 51.) ' Hemihedrons Holohedrons Sphenoidal Pyramidal Trapezohedral mPn mPn [ mPn ] mPn T.I 2 L 2 J 2 mPv mPvc mPv mP* oo Pn vPn \ Pw 1 oo Pn L 2 J ooPoo coPco ooP oo ooP oo mP mP mP mP 2 Hemimorphism. Hemimorphism is observed in the follow- ing classes of the tetragonal system, viz., holohedrons and pyramidal hemihedrons. Tetartohedrism. Although tetartohedral forms are possible in this system, no crystals have been observed with tetartohedrons upon them, consequently they are not discussed. CHAPTER IX. THE ORTHORHOMBIC SYSTEM. Systems Possessing No Principal Plane of Symmetry. As the hexagonal and the tetragonal systems are classed together in consequence of the possession by them of one principal plane of symmetry, so the orthorhombic, the monoclinic, and the triclinic systems may be united into a group characterized by the entire lack of principal planes of symmetry. These systems possess certain analogies which are expressed in part by the names given to their characteristic forms. In each system there are three axes, and these all possess different unities. No two are equiva- lent, hence the symbol of the axes for each system is a : b : c. The Orthorhombic System. The holohedral division of the orthorhombic system includes forms possessing three secondary planes of symmetry. These are per- pendicular to each other, and divide space into eight octants (Fig. 159). Its forms possess in addition three axes of binary symmetry, perpendic- ular to the three planes of symmetry, and a center of symmetry (Figs. 160 and 161). Axes of the System. The lines chosen as the axes of the system are the lines of intersection of the three planes of symmetry in the holohedrons. They are consequently three lines at right angles to each other at a com- mon point (Fig. 162). Since there is no plane of symmetry situated between any two of the axes, no two of them can be equivalent, hence no two can possess the same unities. The two ends of each axis, however, are equivalent since they are separated in each case by one of the planes of symmetry, hence, except in cases of hemimorphism, the two ends of each 102 \ FIG. 159. Distribution of planes of symmetry in ortho- rhombic holohedrons. THE ORTHORHOMBIC SYSTEM 103 of the center of axis must be similarly terminated. Because symmetry, the forms must have parallel sides. Designation of the Axes. Since there is no principal plane of symmetry, there is no one axis that differs from the other axes in any essential particular. Any one of the axes may be selected as the vertical axis, when the other two become the lateral axes. One of these is longer than the other. The longer axis is (Fig. 162) designated as the macroaxis, and the shorter as the brachyaxis. FIG. 1 60. Model showing dis- tribution of symmetry elements in orthorhombic holohedrons. FIG. 161. Symmetry ele- ments of orthorhombic holohe- drons, projected on the plane of the lateral axes. In the scheme of the axes, the brachyaxis runs from front to back is the a axis; the macroaxis from left to right is the b axis; and the vertical axis from below to above is the c axis. In the symbols representing the axes and the crystal planes, the sign ~ always refers to the brachyaxis, and the sign ~~ to the macroaxis. The symbol of the axes thus becomes a : b : c'. Groundform and Axial Ratio. The groundform in this system is composed of planes cutting the three axes at different distances, which are assumed as the unity (Fig. 163) distances on the several axes. The entire form consists of eight planes, one in each octant, constituting a bipyramid whose faces are unequi- lateral triangles. In cross section the pyramid is a rhomb (Fig. 164) and not a square (Fig. 165) as in the case of the tetragonal pyramid. A different groundform is chosen for each substance crystal- lizing in the system. From this the relative lengths of the unities on the c and the a axes are determined in terms of the unity on b, and these are taken as the unities to which all of 104 GEOMETRICAL CRYSTALLOGRAPHY the planes occurring on the crystals of this substance are referred. The relative values of the unities on the three axes constitute the axial ratio, which in this system contains one more term than the axial ratio in the tetragonal and in the hexagonal systems. b- FIG. 162. Axes of the orthorhombic system. FIG. 163. Groundform in the or- thorhombic system. P or in. The value of the axial ratio is so characteristic for each mineral that it is always given under the name of the mineral in the larger text-books on mineralogy. The axial ratio for olivine [(MgFe) 2 SiOJ, for instance, is a : b ^=.4657 : i : .5865. This means that the plane which cuts the three axes at the unities FIG. 164. FIG. 165. Cross sections of groundforms in orthorhombic (Fig. 164) and tetragonal (Fig. 165) systems. intercepts the a axis at .4657 times the distance from the inter- section of the axes as that at which it cuts the b axis; and the c axis at . 5865 times this distance. The plane that cuts the three axes at .9314: i : 1.7595, respectively, is the plane 2a : b : y. THE ORTHORHOMBIC SYSTEM 105 HOLOHEDRAL DIVISION. (Orthorhombic Bipyramidal Class.) The Most General Form. The most general form in the system is composed of planes cutting the three axes at different distances, which do not bear to each other the relations of the unity distances. The symbol of one of its planes may be either na : b : me or a : nb : me. These symbols do not represent dif- ferent planes on the same form, as corresponding symbols do in the tetragonal system, because symmetry does not demand the presence of both, when either is present. The planes occur in- dependently. The plane na : b : me requires the presence of na : b : me, and also na : b : me and na : b : me, above the horizontal plane of symmetry, and the presence of four cor- responding planes below the symmetry plane; but none of these planes demands the presence of a plane cutting the axis b at nb. The plane a : nb : me likewise demands the presence of seven more similar planes, but its presence on a crystal does not demand the presence of any plane cutting the a axis at n. Consequently the most general form in the orthorhombic system is an eight- faced bipyramid. Three Series of Forms. As in the tetragonal system there are three series of pyramids and prisms in the orthorhombic sys- tem. These are named with reference to their relations to the lateral axes. // is customary to regard the shorter distance at which a plane cuts the two lateral axes unity and to call the series by the name of the axis upon which the parameter is not unity, provided the two lateral axes are not both cut at unity. There are thus three series of forms the groundform or unit series, the brachy series, and the macro series. The Unit Pyramids and Prisms. The unit pyramids and unit prisms are composed of planes cutting the two lateral axes at unity. These two axes are cut at different distances, but they are distances that bear to each other the same ratio as do the unities on the axes. The intercept on the vertical axis may be unity or any multiple or small fraction of this. The unit pyramids, called also the orthorhombic bipyramids, IO6 GEOMETRICAL CRYSTALLOGRAPHY are composed of 8 unequilateral triangles whose apices are at the terminations of the vertical axis. A cross section through the lateral axes would exhibit a rhomb, whose diagonals would represent these axes. The symbol of one of its planes is a : b : me, and the symbol of the form is P (m) or mP (hhl). The groundform is that member of this series of pyramids in which the intercept on the c axis is unity. For instance, if the axial ratio of a given substance is .4657 : i : .5865, and its crystals contain planes whose intercepts on the a and b axes bear the relation .4657 : i, these planes belong to the unit series of pyramids. If the relation of the intercepts on the b and c axes FIG. 166. Groundform and two macro pyramids in the orthorhombic system, P, Pn and P>?. is as i : . 5865 the pyramid is P, or the groundform. If the ratio between the intercepts on b and c are as i : 1.7595, tne symbol of the form is 3? (i.e., mP). When m becomes oo the planes of the forrn are all parallel to c, and there results the unit prism whose symbol is ooP or no. This form differs from the corresponding tetragonal prism in" its cross section. The Macroseries. This series consists of forms whose planes cut the brachyaxis (a) at unity and the macroaxis (b) at some distance other than unity. The most general symbol of a plane belonging to this series is a : mb : me. The form composed of planes of this kind is a macro- bipyramid which can be distinguished from the unit bipyramid only when the unity on b is known. Its symbol is mPn or (hkl). THE ORTHORHOMBIC SYSTEM 107 The macro-mark over the n signifies that this parameter refers to the macroaxis. Fig. 166 represents the groundform and two macropyramids with different intercepts on b. It sometimes happens that two crystallographers working on the same crystal choose different pyramids for the groundform and thus obtain different axial ratios for the same substance. But in these cases the two groundforms chosen may bear to each other the relations of unit pyramids to macropyramids or_brachy- pyramids. In the example of olivine cited above (p. 104) the axial ratio accepted is .4657 : i : .5865. The groundform P cuts the three axes at the relative distances indicated. The plane a : 26 : c, cuts these axes at .4657 : 2 : .5865: It is easily conceivable that some crystal- lographer might prefer to use as the groundform the pyramid com- posed of planes cutting the three axes at these distances. If so, his choice of axial ratios would be: .23285 : i : .29325. The original groundform would then be a brachypyramid, 20, : b : 20 or 2?2. Thus in order to distinguish between macropyramids, brachypyramids, and unit pyramids on crystals it is : necessary to know what axial ratio is accepted by crystallographers- as their standard of reference for its forms. When m in the symbol mPn becomes unity a macropyramid results that differs from mPn in the inclination of its faces to the c axis. Its symbol is Pw, or (hlh). When m becomes oo we have the form ooPw, or (hko) which is the macroprism. Figure 167 shows the relation between the unit and one of the macroprisms. The Brachyseries. In addition to the macroseries there is also in this system a series of brachypyramids (Fig. 168) and prisms, which differs from the macroseries in the fact that the lateral parameter which is not unity applies to the brachyaxis. The symbols of the brachyseries are distinguished from those FIG. 167. The unit prism and a macroprism in the orthorhombic system, oo P (no) and ooPw (hko). 108 GEOMETRICAL CRYSTALLOGRAPHY of the macroseries by the use of the mark over the parameter that refers to the brachyaxis. Pw, mPn, (khl), etc., are brachy- pyramids, while ooPw (kho) is the brachy prism. Combinations. In combination the forms of the unit series, the macroseries, and the brachyseries are not difficult to dis- tinguish when they occur together, provided the unit forms can be recognized. The macroforms occur at the terminations of the brachyaxis and the brachyforms at the termination of the macroaxis. When the forms of either series occur alone, how- Fio. 1 68. Groundform and two brachy bipyramids in the orthorhombic system, P, Pw and P'. ever, they can be recognized only by the measurement of their interfacial angles and the calculation from these of the parameters on the lateral axes (see Fig. 174). Domes. There are other classes of prisms in this system that have been given the distinctive name domes. These consist of planes that are parallel to one of the lateral axes, while they intercept the other lateral axis and the vertical axis at certain definite distances. When parallel to the macroaxis, the forms produced by them are called macrodomes (Fig. 169); when parallel to the brachyaxis, the forms are the brachy domes (Fig. In writing the symbols of the domes the parameter on the lateral axis which is not is made unity. Thus the symbols of planes of the macrodomes are a : oo '< : c, or a : 006: me, accord- ing as the c axis is cut at unity or at some other distance. When m becomes , the plane is called a pinacoid. (See next section.) THE ORTHORHOMBIC SYSTEM I0 9 The symbols of the macrodomes are P oo, mP oo, (hoi), and of the brachy domes, P <, mP oo, (ohl). Each dome consists of four faces, uniting in pairs at the terminations of the vertical axis. The macrodomes are in the angles between the vertical and the brachyaxis (Fig. 169), and the brachydomes in the angles between the vertical and the macroaxis (Fig. 170). Pinacoids. The pinacoids embrace those forms with planes parallel to two axes at the same time. Each form consists of a pair of parallel planes. When these are parallel to the two lateral FIG. 169. Two ortho- rhombic macrodomes, P oo and wPcc , 101 and hoi. FiG. 170. Two ortho- rhombic brachydomes, Po and wPco or on and ohl. axes they constitute the basal plane or the basal pinacoid, similar to the corresponding form in the tetragonal system. Its symbol is written oP or (ooi). When the planes of the fo'rm are parallel to the vertical and the macroaxis, the form is termed the macropinacoid, and when parallel to the vertical and the brachyaxis it is called the brachy- pinacoid. The symbol of the former is ocPoo(ioo), and of the latterooPoo, or oio (Fig. 171). Closed Forms. It will be noted that the only forms that will completely enclose space in the orthorhombic system are the bipyramids. No prism, pinacoid, or dome will alone enclose space. All are open forms. Consequently any crystal that contains on it a plane belonging to any one of these three forms cannot be represented correctly by less than .two symbols, no matter how simple the habit of the crystal. Combinations. From a consideration of the statements no GEOMETRICAL CRYSTALLOGRAPHY made in the preceding paragraph, it is plain that the number of forms present in combination on orthorhombic crystals is usually larger than the number found on tetragonal crystals. Figure 172 represents a combination of ocP(/>),ooP <(&), oP(c), and 2? >($, while figure 173 illustrates a more complicated combination of *P(), >P3(/a oP(c), P<2(g), P(o), i/2P(V), i/3P(0") and 2/3?2 (#) . Figure 1 74 represents a combination sometimes seen on andalusite (Al 2 SiO 5 ), ooP^(a), (&), oP(c), ooP(w), ooP2(/), , Poo(r), P(/>), Poo (^), and 2 P2(&). A very complex ^^ OP ^^| 1 ^'"1 col}* cop as 1 x >|Xf r " x^ FIG. 172.^ Crystal of olivine with FIG. 171. Combination of orthorhombic pinacoids and basal plane. crystal of olivine [(FeMg) SiOj is shown in Fig. 175. It contains the forms ooP^ (a), oP (c), ooP (n), ooP2 (s), ooP3 (r), Poo(^), P^ (h), 2P^ (k), 4 P^ W, P W, i/2P (o), 2P2 (/), and Orthorhombic crystals are frequently elongated in the direc- tion of one axis. This direction is usually taken as the vertical axis of the crystals, and the latter are said to possess a prismatic, a columnar, or an acicular habit (Fig. 172). When dispropor- tionately shortened in the direction of a single axis, this axis is likewise often regarded as a vertical axis, and the crystal is said to be tabular in habit. Hemimorphism (Orthorhombic Pyramidal Class). The hemi- morphic development of forms in this system is of considerable importance. With the disappearance of the planes of the holo- hedrons from one end of an axis there is necessarily the disap- THE ORTHORHOMBIC SYSTEM III pearance of one of the planes of symmetry, the two axes of sym- metry lying in this plane, and the center of symmetry. The ele- ments of symmetry remaining are two planes of symmetry and the axis of binary symmetry coinciding with their intersection. FIG. 173. Crystal of topaz. See text for symbols of planes. FiG. 174. Crystal of andalusite. See text for symbols of planes. This axis is necessarily polar, because of the absence of the plane of symmetry perpendicular to it. It is usually made the vertical axis c. All forms except those parallel to two axes, i.e., the prism, the FIG. 175. Crystal of olivine. See text for symbols of planes. macropinacoid, and the brachypinacoids, now separate into upper and lower halves, either of which may occur alone. Figure 176 represents a crystal of struvite (NH 4 MgPO 4 + Poo P oo 6H 2 O) containing ooP oo(7>); at the upper pole, -u(r), - u(q), 112 GEOMETRICAL CRYSTALLOGRAPHY 4? oo oP j./ \x ^ w (0'); and at the lower pole, / (c) and - / (r f ). 2 22 Figure 177 is a crystal of calamine (Zn 2 (OH) 2 SiO 3 ) with ooP oo (6), 7? 00 POO oo Poo ( a ), ooP(); at the upper pole, w ^ (/), - u (r), 2 2 ^P 00 P 00 oP 2? 2 u (q') , (0) , w (c) ; and at the lower pole, / (0) . 222 2 FIG. 176. Hemimorphic crystal of struvite. See text for symbols of planes. FIG. 177. Hemi- morphic crystal of cala- mine. See text for sym- bols of planes. HEMIHEDRAL DIVISION. Kinds of Hemihedrism. The only forms in the ortho- rhombic system that can yield new hemihedrons are the pyramids. One-half the faces of these forms may be extended in three dif- ferent ways, but only in one case will new forms be derived which will comply with the demand of hemihedrism. Consequently there is in this system but one kind of hemihedrism, and by it only one new type of hemihedral form is produced. Sphenoidal Hemihedrism (Orthorhombic Bisphenoidal Class) . The only hemihedrons in this system are two sphenoids which may be regarded as derived from the various pyramids by the extension of planes in alternate octants. These new forms possess only three axes of binary symmetry. The three planes of symmetry and the center of symmetry have disappeared (Fig. This type of hemihedrism is analogous to the inclined hemi- hedrism of the isometric system and the sphenoidal hemihedrism of the tetragonal system. THE ORTHORHOMBIC SYSTEM 113 Sphenoids. The sphenoids are four-sided closed figures com- posed of four scalene triangles meeting in six interfacial edges, two of which are equal and are different from the other four which are also equal. The vertical axis terminates in the centers of the two equal edges, and the lateral axes in the centers of the other FIG. 178. Diagram ill us 1 rating distribution of sym- metry elements in orthorhombic sphenoidal hemihedrons. FIG. 179. Right orth- orhombicbi sphenoid r P , or hhl. FIG. 180. Left ortho- rhombic bisphenoid, four. The form differs in appearance from the tetragonal sphen- oid from the fact that the edges at the terminations of the vertical axes are not at right angles. The difference between the value of this angle and 90 increases with the difference in the lengths of the two lateral axes. From every pyramid two of these forms are produced. They are enantiomorphous, and are therefore known as right (Fig. 179) 8 GEOMETRICAL CRYSTALLOGRAPHY and left forms (Fig. 180). Their symbols are r (hhl), I , 2 2 r (kht), I (hkl), -(hkl), and ** (khl). FIG. 181 Crystal of epsom salts with and r?o). A combination of ooP(/>) and r (0) as seen on crystals of epsom salts (MgSO 4 +7H 2 O) is illustrated in Fig. 181. CHAPTER X. THE MONOCLINIC SYSTEM. Symmetry of the Holohedrons of the Monoclinic System. Complete forms belonging to the monoclinic system possess but a single plane of symmetry. Consequently monoclinic, holohedrons are bilaterally symmetrical; i.e., they possess two sides that are alike. They possess also an axis of binary sym- metry perpendicular to the plane of symmetry and a center of symmetry (Fig. 182). Figure 183 is a diagrammatic representation of the dis- tribution of the elements of symmetry in the plane of the lateral axes. FIG. 182. Model of monoclinic crystal show- ing distribution of ele- ments of symmetry in holohedrons. FIG 183. Diagram illus- trating distribution of sym- metry elements of monoclinic holohedrons. Axes. The single plane of symmetry in this system deter- mines the position of one line which may be chosen as one of the axes of reference for the planes of monoclinic forms. This line is the axis of symmetry which is perpendicular to the plane of symmetry. The other two axes must necessarily lie in the plane of symmetry, but their directions in this plane are a matter of choice, to be decided in the case of each substance as may be most convenient. The two axes that lie in the plane of symmetry are at right angles to the third axis, but are inclined to each other at some angle Il6 GEOMETRICAL CRYSTALLOGRAPHY other than 90. Their angle of inclination is always designated as the angle /? (Fig. 184). Symmetry demands that the two terminations of the axis that is normal to the plane of symmetry shall be equivalent in all respects. There is, however, nothing in the symmetry of the system which necessitates the equivalency of any two of the three axes or of the opposite ends of the axes in the plane of symmetry. Hence the unities on the three axes are different in value. A model representing the axes of the monoclinic system would be composed of three lines of unequal lengths intersecting at a common point. Two of these must neces- sarily be inclined to each other at FIG. 184. Relation of axes to some angle other than 90, and the plane of symmetry in monoclinic third must be norma } to t h e plane holohedrons. of these. Designation of the Axes. It is customary in studying monoclinic crystals to place them in such a position that the plane of symmetry shall stand vertically. The line normal to this plane then takes the position of the b axis. Of the other two axes, one is made vertical and the other is so placed that it inclines toward the observer. The acute angle /? is thus on the back of the crystal (Fig. 184). As in the other systems, the axis that stands vertically is the axis c or the vertical axis. The axis corresponding to the b axis of the orthorhombic system, i.e., the one that is normal to the plane of symmetry, is the orthoaxis. The third axis the one inclined to the vertical axis is the a axis. It is known as the clinoaxis. The sign used in symbols to designate reference to the b axis is the same as that which designates reference to the macroaxis in the orthorhombic system. The sign used to designate the clinoaxis is \. The symbol for the axes is thus a : b : c. Groundform and Crystallographic Constants. As in the THE MONOCLINIC SYSTEM 117 orthorhombic system, so in the monoclinic system, any form composed of planes cutting the three axes at finite distances may be assumed as the groundform, and from it the lengths of the unities on the three axes may be determined. These unities are expressed in terms of the unity on b as i, and when once deter- mined for a crystal of any substance this ratio is accepted as the axial ratio for all crystals of that substance. In addition to the axial ratio there is one other determination necessary to fix the position of the axes in this system, viz., the inclination of the a to the c axis, or the value of the angle /?. The axial ratio and the value of /3 constitute the crystal- lographic constants. In orthoclase (KAlSi 3 O 8 ) these constants are a : b : ^ = .6585 : i : .5554. =63 56' 46", and in augite (a calcium, magnesium, iron silicate) a : b : c= 1.092 13 : i : .58931. 0=74 10' 9". In practice, although the choice of the lines that shall serve as the a and the c axes is purely arbitrary, it is usual to select those that will yield the simplest symbols for the forms most fre- quently found in combination. One or the other of these axes is made parallel to some prominent plane on the crystal or, better, to some prominent zone of planes. These planes then become pinacoids, domes, or prisms, when they are easy to recognize. HOLOHEDRAL DIVISION. (Prismatic Class.) Pyramids. The most general symbol possible in the system is na : b : me, or a : nb : c. Either represents a plane cutting the three axes at different distances, one of which, the intercept on b or on a, is considered as the unity on this axis. The presence of one of these planes necessitates the presence of a corresponding plane on the opposite side of the plane of symmetry, and the presence of these two demands the presence of two others parallel to the former because of the existence in the system of a center of symmetry. Consequently, in this system the most general form consists of two pairs of planes. If one of the planes occurs in an octant containing the acute angle /?, all other planes of the n8 GEOMETRICAL CRYSTALLOGRAPHY form occur also in acute octants. These are known as the positive forms. If, on the other hand, the planes are all in the obtuse octants, negative forms result. A combination of a positive and a negative form whose planes possess the general symbol na : b : me, gives a bipyramid (Fig. 185) analogous to the bipyramids in the orthorhombic system, except for the fact that it consists of two sets of planes of different shapes. Since the positive and the negative forms each constitute half of this bipyramid, they are termed the positive and the negative hemipyramids. A monoclinic bipyramid com- posed of the planes a : b : c, must thus be represented by two FIG. 185. Combina- tion of positive and negative hemipyramids. FIG. 186. Crystal of ferrous sulphate. symbols: +P and P. Either of these hemipyramids may occur independently of the other, as each is completely holohedral. (See Figs. 186 and 188, p. 120.) Of these hemipyramids there are three different kinds corre- sponding to the three kinds in the orthorhombic system. They are named in accordance with the same principles that determine the naming of the orthorhombic forms. a : b : c = +P(iu) and P(in), unit hemipyramids. a : nb : c= -fPn and Pn, orthohemipyramids. na : b : c= +Pn and Pn, clinohemipyramids. When the parameter on the vertical axis becomes m there result other pyramids that belong to one of these three series, as mP(hhl), mPn(hkl,h>k) and mPn(M ) hk), the orthoprism and Pn(kho, h/), mP(hol), etc.; and the negative orthodomes, P oo, mP GO, etc., the former in the acute /? and the latter in the obtuse ft (see planes r and r' in Fig. 1 86). The clinodomes are parallel to the clinoaxis. These are at the same time in both positive and negative octants, and hence possess no signs. Their symbols are Poo(oW), mP&(ohl), etc. (Plane q in Fig. 186). Pinacoids. The pinacoids are parallel to two of the axes at the same time. Their planes must belong to two octants, hence they possess no signs. The basal pinacoid corresponds to the basal planes of other systems. Its symbol is the conventional oP(ooi). The other two pinacoids are the orthopinacoid and the clino- pinacoid. The former is parallel to the vertical and the ortho- axis, and the latter to the vertical and the clinoaxis. Their 120 GEOMETRICAL CRYSTALLOGRAPHY symbols are, respectively, ooP 00(100) and ooP 00(0 10). Theclino- pinacoid is the plane of symmetry (plane b in Fig. 186 and M in Figs. 187 and 188). Combinations. Combinations of monoclinic forms are more varied than are the combinations of the forms belonging to any system of a lower grade of symmetry. This is due to the fact that there are no closed forms in this system. The simplest crystals must be represented by at least two symbols. In working out the symbols of monoclinic crystals it is abso- lutely necessary first to bring the crystals into the correct conven- tional position. The plane of symmetry must first be discovered, and then must be placed vertically and at the same time parallel FIG. 187. FIG. 188. Crystals of orthoclase withooPoo(M),oP(P), ooP(T),2P5o (), + 2P~o%), and + P(o). to the line of sight of the observer. When held in this position, the b axis runs horizontally from right to left. Its position is fixed by the symmetry of the system. The choice of the other two axes is a matter of convenience in each particular case and is regulated largely by the habit of the crystal. If this possesses planes that may be regarded as pinacoids, these planes serve to fix the posi- tions of the two axes in question. The planes assumed as basal planes determine the position of the clinoaxis. The following illustration taken from Dr. Williams's Elements of Crystallography may serve to make these points clear. On the crystal of iron- vitriol (FeSO 4 +7H 2 O), represented in figure 186, the only plane whose value is absolutely fixed is the plane of symmetry, or clinopinacoid (b). It is customary to make c the THE MONOCLINIC SYSTEM 121 basal pinacoid and p the fundamental prism, whence q becomes a clinodome, o a negative hemipyramid, and r' and r plus and minus hemiorthodomes. We might, however, turn the crystal so as to make c the orthopinacoid, q the prism, and p a clinodome; or we might even make r f the basal pinacoid, and r the orthopina- coid, when o would become the prism, c a hemiorthodome, and p and q both pyramids. Figures 187 and 188 represent two common combinations in this system. The two crystals are very different in habit though FIG. 189. Crystal of real gas. See text for symbols of forms. FIG. 190. Crystal of gypsum, with ooP (/>),ooPoo(ft) and they possess nearly the same forms. Both are crystals of the feldspar known as orthoclase (KAlSi 3 O 8 ). In the figures, M= ocP^c, P = oP, T= ooP, y= + 2P, w = 2?oo, x= +P<*> and o= +P. Figure 189 is a crystal of realgar (As 2 S 2 ) wth oP(P), ooPoo(r), ocP(Jlf), ooP 2 (/), Poo() and +P(s). Figure 190 is a crystal of gypsum (CaSO 4 +2H 2 O) with ocP(/>), ooPoo(&) and HEMIHEDRAL DIVISION. (Domatic Class.) Hemihedrism in the Monoclinic System. By the selection of half the planes of a hemipyramid in such a way that opposite ends of the ortho-axis are equivalently terminated, i.e., if each end is terminated by one plane, the conditions of hemihedrism for 122 GEOMETRICAL CRYSTALLOGRAPHY this system are complied with. The resulting hemihedron still retains the plane of symmetry, but the axis of symmetry and the center of symmetry are gone (Fig. 191). The new form derived from P consists of the upper or the lower half of the hemipyramid, and is known as the tetrapyramid. Each hemipyramid, therefore, yields two hemihedrons, which are designated the upper and the lower tetrapyramids. Their sym- P P 7 P P 7 mP mPn mPn bolsareH u, H /, u, 1, u.l, u.l, u.L, etc. 22222 2 2 FIG. 191. Diagram illus- trating distribution of sym- metry elements in monoclinic hemihedrons. FIG. 192. Crystal cf K 2 S 4 O e showing hemihedral forms. See text for symbols of forms. In the same way the orthodomes yield upper and lower tetraorthodomes, the clinodomes yield upper and lower hemidino- domes, and the prisms front and rear hemiprisms. The basal plane yields an upper and a lower form, and the orthopinacoid a front and a rear form. oP co P GO ooPoo Figure 192 is a combination of u (c), j (a), r 22 2 ooP ooP Poo P2 (a behind), f(m) and r (m behind) , u (q), / (u), and 22 22 p H l(o), on crystals of potassium tetrathionate (K 2 S 4 O 6 ). Hemimorphism (Sphenoidal Class). If one-half the planes of the hemipyramid are grouped at one end of the orthoaxis, the other half being absent, the plane of symmetry disappears, the THE MONOCLINIC SYSTEM 123 axis of symmetry becomes polar, and the center of symmetry no longer exists. The resulting form has but one element of sym- metry which is the polar axis of binary symmetry (Fig. 193) and it is hemimorphic along the orthoaxis. FIG. 193. Diagram illstrating symmetry elements in monoclinic hemimorphs. Each hemipyramid thus breaks up into two tetrapyramids, one / P P\ of which is at the right end of the axis! +r,r] and the V 2 2/ / P P\ other at its left end ( +/-, -A). 2 2 FIG. 194. FIG. 195. Right- (Fig. 104) and left- (Fig. 195) handed crystals of tartaric acid with ooPoo )' r ^j~ (P)t^f (P'\ oP(c), P36 (r), r?~ (q in Fig. 194) and /^- (q in Fig. 195). The prisms, the clino-domes, and the clino-pinacoids likewise may give rise to right and left forms, but the forms with planes that are parallel to the &-axis can yield no new hemimorphs. In all cases the corresponding right and left forms are enantio- morphous. On the crystals of tartaric acid (C 4 H 6 O 6 ), represented in figures 194 and 195, the plane p is a plane of the prism at the right 124 GEOMETRICAL CRYSTALLOGRAPHY ooP end of the axis, or is r , and p' a similar plane at its left end or ooP P io / . The plane q in figure 195 is / and q in figure 194 P 00 r . The former is a left-handed crystal and the latter a right- 2 handed crystal, so called because their solutions turn the plane of polarization of light to the left and the right, respectively. CHAPTER XL THE TRICLINIC SYSTEM. (Pinacoidal Class.) Symmetry of the Triclinic System. The forms belonging to the holohedral division of the triclinic system possess no planes of symmetry. There is, however, a center of symmetry. Each form, therefore, consists simply of a pair of parallel faces. Every pair of planes on triclinic crystals must therefore be represented by a different symbol. Axes. As there are no planes of symmetry in this system, the axes to which the forms are referred must be chosen arbi- trarily. In the systems possessing three or more planes of symmetry all the axes are determined by the symmetry. In the monoclinic system there is only one plane of symmetry, hence the position of one axis only is determined. In the triclinic system there are no planes of symmetry. Consequently, the position of no axis is fixed. Convenience alone determines the choice of axes. FlG - ^6 -Axes of the triclinic system. Practically, certain planes are assumed as prisms or pinacoids, and lines parallel to these are made the axes. Each substance crystallizing in this system has a different set of axes, which, however, when once established become the axes for all crystals of that substance. The axes in the system differ, not merely in their unit lengths, but also in their inclinations. Each is inclined to the other two at the point of their common intersection. A model of the unit lengths of the triclinic axes would show three lines of different lengths intersecting each other at a common point, and obliquely inclined to one another (Fig. 196). Designation of the Axes. When the axes of a crystal are I2 5 126 GEOMETRICAL CRYSTALLOGRAPHY once decided upon, one of them is held vertically as the vertical axis. Of the other two, the longer is made the macroaxis, b, and the shorter the brachyaxis, a. The macroaxis is made to incline downward toward the right and the brachyaxis down- ward toward the front. The signs used to designate the axis are the same as in the orthorhombic system. Thus the symbol of the axes in this system is a : b : c. Groundform and Crystallographic Constants. The groundform in this system consists of two parallel planes in opposite octants, each cutting the axes at the same distances, which, however, are different on the different axes. The inter- ' cepts of these planes are the unity lengths for the substance on which the form occurs. As in the other systems with unequal axes, the unities on a and c are measured in terms of the unity on b. This axial ratio varies for different substances. In order that the position of the axes be fixed it is necessary to know their angles of inclination to one another, just as in order to describe the positions of the a and the c axes in the monoclinic system it is necessary to specify the value of the angle /?. In the triclinic system each of the axes is inclined to the other two, hence it becomes necessary to determine the values of three - angles. These are designated as the angles a, /?, and r(see Fig. 196). The angle a is included between the vertical and the k macroaxes. The angle /? between the axes c and a, and the angle 7- between the axes a and b. The crystallographic constants in this system thus comprise six elements; viz.: the lengths of the three axes, and the angles a, /?, r- For jilbite (NaAlSi 3 O 8 ) the crystallographic constants are a : b : =.6338 : i : . 5577, and a =94 3', j0=n6 28 5/6', r = 88 8 2/3'. Pyramids. The pyramids of the triclinic system differ but little in the nature of their planes from the pyramids in the mono- clinic system. Their planes cut the three axes at different dis- tances. If these distances are those that are assumed as the unity distances, the pyramid is a unit pyramid. If the intercept on one of the lateral axes is greater than unity on this axis when it is unity on the other, the pyramid is a br achy pyramid or a macro- THE TRICLINIC SYSTEM 127 pyramid according as the greater intercept is on the macroaxis or on the brachyaxis. Since there is no plane of symmetry in the triclinic system, the existence on a crystal of one pyramidal plane necessitates the presence of no other corresponding plane except that which is parallel to it in the opposite octant, this being demanded by the center of symmetry. Consequently, a complete geometrical bi- pyramid comprises four pairs of parallel planes, each pair of which must be represented by a distinct symbol (Fig. 197). As each pair of planes constitutes one-quarter of a complete pyramid, it is known as a tetrapyramid, and is represented in the unit series by P with an accent written near (Fig. 197) it in a position correspond- ing to the position of the front octant in which the plane occurs. Thus P' repre- sents the form, one of whose planes is in the upper right-hand octant in front. P/ the form whose front plane is in the , . i, , , -r, ,, - FIG. 107. The four triclinic lower right-hand octant, ,P the form tetrapytamids. whose front plane is in the lower left- hand octant, and 'P the form one of whose planes is in the upper left-hand octant. /P' n and /P/ n are the corresponding macrotetrapyramids and brachytetrapyramids. The indices of these forms are the same as those of the planes on the front of the pyramids of the tetragonal, orthorhombic, and monoclinic systems. Prisms. Since the prisms are composed of planes parallel to the vertical axis, it necessarily follows that these planes are at the same time in two octants. Each prism consists of two paral- lel planes opposite each other. There are thus in this system two hemiprisms whose symbols are, respectively, QO P,, and oo , P f or the unit series, oo ~P',n and oo 'Pn for the mac rohemi prisms and P',n and GC 'Pn for the brachyhemiprisms. The forms oo P,, GO P, n and QO P, n have planes on the right-hand side of the front of the crystal, and QO ,P, oo fjj*n and oo 'Pn have front left-hand planes. The form oo P,, and the corresponding macro- and brachy-forms are derived from P' or from P/ by the change of the intercept on c tooo, and the 128 GEOMETRICAL CRYSTALLOGRAPHY forms oo 'P, etc., from ,P or from 'P in the same manner. The corresponding indices of the hemiprisms are: oo P', iio;ocP'w, Mo;ocP,n, kho oc'P, iio;',P2(l) and P'(s). Figure 201 represents a crystal of anorthite (CaAl 2 Si 2 O 8 ). It is much more complicated than the blue vitriol crystal as it contains the forms oP(P); ooP^(M); ooP^/); ooIPfT); ooP^f/); 00^3(2); 9 130 GEOMETRICAL CRYSTALLOGRAPHY F(iii); 'P(a); P, (*); ,P (#) ; 4 P< 2 (v) ; T'^(0; 2,P, (y); ,P' * () ; 2,P' 5 (r) and T, (). Comparison of the Systems. The main differences between the crystallographic systems are conditioned by differences in the symmetry of the holohedrons belonging to them. The differences are expressed primarily in the character of the axes chosen for each system. For that system possessing the highest grade of symmetry the unities on all the axes are of equal length and the axes themselves are fixed in position, being perpendicular to one another. In the system possessing the next lower grade of symmetry there are four axes, one of which is of different length from the other three. All are fixed in position. The three axes of the same length are inclined to each other at angles of 60 and are perpendicular to the fourth axis. The system of the next lower grade has three axes, one of which is different in length from the other two which are equal. All are at right angles to one another. These three systems are all characterized by the possession of at least one principal plane of symmetry. Of the systems having no principal plane of symmetry, that of the highest grade of symmetry possesses three axes of different lengths perpendicular to one another. The next system, which has only one symmetry plane, has only one of its axes fixed in position. This is perpendicular to the plane of symmetry. The other two axes are in the symmetry plane, but their positions in this plane are not fixed. Lines lying within this plane are arbitrarily chosen to serve as axes. The axes, therefore, are three unequal lines; one at right angles to the plane of the other two. The system with the lowest grade of symmetry has no axes fixed for it. They are chosen arbitrarily and must consist of three unequal lines inclined to one another at angles other than right angles. The symbols for the axes become more and more complex as we pass from the systems of higher to lower grade, and more variable factors are concerned in them. These facts may be expressed by writing the symbols as follows, omitting angles of 90: THE TRICLINIC SYSTEM 131 Systems. Axes Isometric a Hexagonal a : c Tetragonal a : c Orthorhombic a : b : c Monoclinic ; . a : b : c ; /? Triclinic a : b : c ; a, ft 7-. CHAPTER XII. CRYSTAL IMPERFECTIONS. Ideal Forms. Occasionally a crystal occurs which possesses the regularity of a well-made model. All the faces belonging to the same form have the same size. They are all equally developed and are all at equal distances from the center. These are ideal forms like those represented by most of the figures in this book. Usually, however, the regular and sym- metrical growth of a crystal has been interfered with by some external agency or condition unfavorable to the symmetrical development of the ideal form. Moreover, it frequently happens that the faces of crystals are not perfectly plane as they are FIG. 202. FIG. 203. FIG. 204. Symmetrical and distorted cubes with columnar (Fig. 203) and tabular (Fig. 204) habits. assumed to be in crystallographic discussions, and occasionally the values of the interfacial angles vary from the calculated values for the ideal form. These are all imperfections in the crystals. They may be classified as imperfections in symmetrical develop- ment, imperfections in faces, and imperfections in angles. Distorted Crystals. The unequal rapidity of crystal growth in different directions often results in the elongation of the crystal in certain directions, and the consequent increase in the size of some of its planes at the expense of others. This process when carried to excess actually crowds out some of the faces that should 132 CRYSTAL IMPERFECTIONS 133 appear on the crystal, apparently destroys its symmetry, and produces bodies with the shapes characteristic of crystals of a lower grade of symmetry. The cube may grow much faster in the direction of one axis than in the direction of the others, and thus may simulate a tetragonal prism and basal plane, or it may grow rapidly in two directions, producing a tabular crystal (Figs. 202, 203, and 204). The dodecahedron of the isometric system FIG. 205. FIG. 206. FIG. 207. Dodecahedron distorted in such a way as to appear to possess tetragonal (Fig. 205), hexagonal (Fig. 206), and orthorhombic (Fig. 207) symmetry. may by elongation in different directions come to resemble a tetragonal crystal (Fig. 205), an hexagonal one (Fig. 206), or an orthorhombic one (Fig. 207). By elongation in two directions, that is, by flattening, tabular crystals may be produced which at the first glance do not resemble in the slightest degree the forms from which they were derived. Figure 208, for instance, illustrates an octahedron that has been flattened in the direction of a line passing through the centers of opposite octahedral faces. Nevertheless, however much crystals may be distorted in this way, their interfacial angles always have the values possessed by the cor- responding ideal forms and their physical properties are the same. Moreover, by im- agining the corresponding faces on all sides FI G. 208. Flattened of the crystal to be moved toward its center, the form may be restored to the shape and symmetry properly belonging to it. Habit of Crystals. The habit of a crystal, or its general shape, depends largely upon the character of the distortion it has suffered. It is often so characteristic a feature of different sub- stances that a number of terms have been invented for trie purpose of describing it. 134 GEOMETRICAL CRYSTALLOGRAPHY When a crystal grows much more rapidly in one direction than in others, and consequently is elongated in a single direction, it is said to have an acicular habit or to be acicular, when its cross section is very small; to be prismatic or columnar, when its cross section is greater. Quartz crystals are columnar in the direction of the c axis, orthoclase in the direction of the a axis, and epidote in the direction of the b axis. Figures 209, 210, and 211 represent crystals of these three substances. When the elongation is in the direction of two crystallographic axes or, more properly, when the crystal is most largely developed laterally, it is tabular, as are many crystals of mica. oP 031*00 FIG. 209. FIG. 210. FIG. 211. FIG. 209. Columnar crystal of quartz elongated parallel to c. FIG. 210. Columnar crystal of feldspar elongated parallel to a. FIG. 211. Columnar crystal of epidote elongated parallel to &. Other terms besides those defined are used in the descriptions of the habits of crystals, but they are self-explanatory and so need no special definition. Deformed Crystals. Distortions of another kind often produce very peculiar results, such as twisted or curved crystals, bent crystals, rounded crystals, etc., all of which are caused either by the prevalence of very unusual conditions during growth or by the action upon the crystal of mechanical forces after its formation. Many crystals imbedded in rocks are bent, and often they are cracked or shattered. Distortions often affect not only the shapes of the crystals, but often their interfacial angles as well. Figure 212 is a picture of a curved hornblende CRYSTAL IMPERFECTIONS crystal in a rock from near Marquette in Michigan. The rock is magnified 30 diameters. Imperfections in Crystal Planes. Although theoretically crystal faces are plane surfaces, practically a very large proportion FIG. 212. Curved crystal of hornblende in rock. (After G. H. Williams.} of the crystal planes found in nature are not even. The unevenness may be due to striations, curvature, corrosion, or irregularity in growth. Striations. Striations are fine parallel lines that sometimes extend across all the faces of a crystal and at other times are FIG. 213. Striated quartz crystal. FIG. 214. Striated crystal of pyrite. FIG. 215. Striated crystal of plagioclase. limited to the planes belonging to a single form. These may originate in the oscillatory combination of two distinct forms, as the horizontal striations on the prismatic faces of many quartz crystals (Fig. 213), which are produced by the oscillatory i 3 6 GEOMETRICAL CRYSTALLOGRAPHY combination of ooR and R, or on cubes of pyrite (Fig. 214) due [wOool to combinations of oo O oo and - . They may also be due to polysynthetic twinning, as in the case of striations on the prismatic, the basal, and the macrodome faces of plagioclase crystals (Fig. 215), when the twinning lamellae are so small that they can with difficulty be recognized. (See p. 155-156.) Curvature. Many substances, like diamond (C) (Fig. 216), dolomite ((CaMg)CO 3 ), etc., are so frequently bounded by curved faces that their occurrence must be ascribed to some characteristic property of the molecules of which these bodies are composed. FIG. 216. Crystal of diamond with curved faces. FIG. 217. Natural etched figures on diamond crystal. (After Tschermak.) Corrosion and Etched Figures. After a crystal has been formed with perfect planes, solutions may attack it and dissolve portions of the faces, rounding their edges and so causing the plane to become curved, or else pitting them with little hollows, known as etched figures. The etched figures vary in shape in different minerals and on different planes of the same mineral, their shapes always being governed by the symmetry of the plane on which they occur. Those on the cubic faces of diamond (Fig. 217) are hopper-shaped. Imperfections Caused by Irregularities in Growth. The irregularities developed during the growth of a crystal are too numerous to be specified. Often an obstruction met with by the crystal will become imbedded in it and so cause one or more of its planes to become distorted, or it may impress its shape upon CRYSTAL IMPERFECTIONS 137 a plane with which it comes in contact and in this way destroy its perfection of surface. Or a crystal may grow rapidly, forming its edges first and building up a skeleton, which may not become entirely filled up. In place of a crystal face there may thus result a reversed pyramid, as in the so-called hopper-shaped crystals of salt, crystals of cuprite (CuO) (Fig. 218), or galena (PbS), and in metallic bismuth (Fig. 219). Symmetry of Imperfections. Although the imperfections in crystal faces tend to destroy their ideal development, these imperfections are nevertheless governed in a great measure by the symmetry of the crystal form. All cry stallo graphically equivalent FIG. 218. Skeleton crystal of cuprite. FIG. 219. Portion of skeleton crystal (cube) of bismuth. planes are similarly affected. The symmetry of the imperfections on forms in combinations or even in apparently simple forms may often serve to aid in the correct determination of their sym- metry. In this way certain forms that appear to be holphedral are shown to be in reality hemihedrons that are not geometric- ally distinguishable from the holohedrons from which they are derived. The striations on cubes of pyrite (FeS 2 ) (see Fig. 214) are often arranged symmetrically with respect to the three principal planes of symmetry passing through the cube, and with respect to these only. The cube is thus symmetrical in the same way as are the hemihedrons derived by parallel hemihe-drism, hence the form may be regarded as a hemihedral cube derived from the holo- 138 GEOMETRICAL CRYSTALLOGRAPHY FIG. 220. Crystal showing distribution factions. Sp) of galena, of imper- hedral cube by the parallel method of hemihedrism. When in combination with distinctive hemihedrons, these are always either the diploid or the pyritoid, the characteristic hemihedrons of parallel hemihedrism, and never forms derived by the gyroidal or the inclined methods of hemihedrism. Again, on the mineral sphalerite (ZnS) there frequently occur eight triangular faces distributed like the eight faces of an octa- hedron (Fig. 220). But of these, four are smooth except for a few striations, and four are rough, a smooth and a rough face alternat- ing. The rough faces correspond to the faces of a tetrahedron and the smooth ones to those of a corre- sponding tetrahedron with an op- posite sign. If the former be con- sidered + , the latter is . 2 2 These forms are inclined hemihe- drons, hence we should expect all other forms on this same mineral to be inclined hemihedrons with characteristic forms, or holohedrons that do not yield new forms by the inclined method of hemihedrism. As a matter of fact, these are the only forms that do occur in the mineral. Variations in Crystal Angles. The imperfections in the planes on crystals do not often affect the values of their interfacial angles. When these vary the variations are usually due to a difference in the chemical composition of the minerals whose angles are being compared, to differences in the temperature at which the crystals are measured, or to some mechanical force acting upon the crystal from without. The variations are always slight and, except in the case where mechanical force is concerned, the variations only emphasize the fact that the interfacial angles on crystals are their most characteristic features. Impurities within Crystals. Ideal crystals consist of homo- geneous matter possessing the same chemical composition through- out and having the color of the pure substance that composes their mass. Departures from this ideal purity are frequently met with. CRYSTAL IMPERFECTIONS 139 The impurities may consist of a dilute coloring matter, or pigment, which is present in such small quantity as to be inde- terminable, but nevertheless in sufficient quantity to give a de- cided color to the crystals. Pure quartz (SiO 2 ) is colorless, smoky quartz is gray or black, but the coloring matter in it is present in such small quantity that its true nature is not certainly known. Sometimes the coloring matter is evenly distributed throughout the entire crystal, sometimes it is regularly distributed in concen- tric layers, called zones, and sometimes it is irregularly distributed without respect to crystallographic directions. Tourmaline crystals from Hebron and Mount Mica, Maine, very often have a pink nucleus sur- rounded by zones of different shades of green. Figure 221 represents a cross sec- tion through such a crystal. F[G 22I _ Cross sec . The other impurities in crystals may tion of crystal of tour- , , , , . , . ,. maline exhibiting zonal be classed together as inclusions. They can arrangement of color, usually be distinguished from the mineral substance in which they are imbedded, if not by the eye, at least under the microscope. They comprise gas, fluid, glass, crystal, and unindividualized inclusions. Gas inclusions are found in crystals of both aqueous and igneous origin. They appear as little cavities of different shapes, in some cases scattered promiscuously through the crystal sub- stance; in others arranged zonally. When the cavities have the shapes of the crystal they are known as negative crystals. Their contents may be air, carbon dioxide (CO 2 ), marsh gas (CH 4 ), or some other hydrocarbon, sulphur dioxide (SO 2 ), or some simple gas. Fluid Inclusions. When the pores are filled with liquid they are known as fluid cavities. Like the gas inclusions these may be of different shapes, and, like them, they may be arranged regularly or irregularly within the crystal substance. The fluid enclosed in the cavities may be water, liquid carbon dioxide, some other liquid gas, petroleum, or some dilute salt solution. Often the liquid encloses a bubble of gas (Fig. 222) which by its movements 140 GEOMETRICAL CRYSTALLOGRAPHY serves to distinguish the fluid from the glass inclusion. Figure 223 is a magnified view of gas and liquid inclusions in quartz. In a few cases two fluids which do not mix are enclosed in the same cavity in which case that one whose surface tension is greatest occupies the center of the cavity, while the other surrounds it as an envelope (Fig. 224). When liquid and gas are both enclosed in the same cavity, the size of the gas bubble may be made to change by increasing or diminishing the temperature, and at the proper temperature may be made to disappear entirely. By noting the temperature at which the bubble disappears, i.e., the FIG. 222. Liquid inclusions in quartz. Slightly magnified. FIG. 223.- Liquid inclusions in quartz. Greatly magnified. temperature at which there is no distinction between the liquid and the gas, the nature of the inclusion may sometimes be determined. These phenomena are supposed to show that the substance was crystallized under high pressure, and that while growing it surrounded a portion of the mother liquor from which it was separating. Glass inclusions occur in crystals that have solidified from a molten mass and have enclosed a portion of this mass during growth, or in those which have since their formation been heated so high as to melt foreign substances contained within them. These inclusions also often contain bubbles, but in this case they are immovable. The existence of two bubbles in the same inclusion is proof that the cavity is filled with glass. CRYSTAL IMPERFECTIONS 141 Crystal inclusions. These are minute crystals caught up and enclosed in larger crystals during growth, or minute crystals formed by the decomposition of the latter. FIG. 224. Thin section of smoky quartz showing inclusions consisting of central gas bubble, surrounded by two liquids that do not mix. The outer liquid is probably water and inner liquid and gas bubble carbon dioxide. Magnified about 150 diam. (After Rosenbusch.) When a crystal of orthoclase solidifies from a solution in which there occur already formed tiny crystals of apatite, some of these small crystals may become embedded in the orthoclase as inclu- sions. On the other hand, when orthoclase under the influence FIG. 225. Section of leucite crystal showing zonal arrangement of minute inclusions. (After Tschermak.} of chemical agents begins to change into kaolin, the first stages of the alteration are seen in the presence of tiny scales of kaolin scattered through the mass of the orthoclase. In some crystals the inclusions are so arranged as to produce an iridescence in the 142 GEOMETRICAL CRYSTALLOGRAPHY enclosing substance, as in the case of labradorite, bronzite, etc. In others they are arranged zonally, as in leucite (Fig. 225) . When so small as to baffle attempts to identify them as definite mineral species, the inclusions are known as crystallites; when large enough to be identified, they are referred to as microlites. The inclusions in crystals can best be studied in very thin slices. When these are viewed under the microscope it is found that very few crystals are free from impurities of many kinds. Nearly all contain either liquid, gas, or mineral inclusions. The former are especially abundant even in crystals occurring in the hardest rocks. Often in a single grain (Fig. 222) of the quartz in a granite will be seen hundreds of tiny pores nearly filled with liquid. In many of these are little bubbles that move slowly through the liquid mass or dance rapidly, as the case may be, moving inces- santly, in consequence, probably, of the slight changes of tempera- ture to which the substance containing them is subjected. Some- .times, but not as commonly, a liquid inclusion will contain a little crystal of some salt, which may be made to dissolve by warming the specimen or to grow larger by cooling it. The quantity of liquid enclosed in some minerals is very large, reaching, it is said, in the case of the constituents of certain rocks as much as i . 8 per cent, of their volume. Often the liquid is really a condensed gas, most commonly liquid carbon dioxide, methane, or nitrogen. The volume of gases obtained from en- closures in quartz crystals from Poretta, Italy, amounted in one instance to . 03 per cent, of the volume of the quartz. The total quantity of gas that has been obtained from certain other crystals is much greater, reaching 13 per cent, by volume in some cases, but much of this was present in some other form than as enclosures in cavities. CHAPTER XIII. CRYSTAL AGGREGATES. Crystal Individuals and Crystal Aggregates. Thus far only crystal individuals have been discussed, since these most nearly exhibit the ideal forms demanded by symmetry. Rarely, however, are crystal individuals complete. There is usually lacking some one or more of their faces where the crystal was attached during its growth. Individual crystals that exhibit only an occasional face and crystallized masses that possess forms due to influences acting from without the crystallizing body (allotriomorphic forms) are distinguished as crystal grains. Neither crystal individuals nor crystal grains commonly occur isolated. They are usually grouped together irregularly. Often, however, several crystals or parts of crystals are grouped in parallel positions, or in such a way that the different individuals are separated from one another by a plane about which they are ' symmetrically disposed. The former of these definite groupings are known as parallel growths and the latter as twinned crystals. l The general term "aggregate" is given to the irregular groupings. Irregular Aggregates. Irregular aggregates may consist of grains of the same mineral irregularly grouped, as are the particles of calcite in a coarse-grained limestone (Fig. 226), or they may be composed of grains of several substances, as is the aggregate of orthoclase (KAlSi 3 O 8 ), quartz (SiO 2 ), and mica known as granite (Fig. 227). Such aggregates as these, where each grain is completely or almost completely bounded by other grains, are distinguished as crystalline aggregates. When the grains in an irregular aggregate are partially bounded by their own planes, the aggregate is a crystal aggregate; and when the individual grains approach in form the character of crystal individuals, the aggregate is a crystal group. 144 GEOMETRICAL CRYSTALLOGRAPHY Figure 227 is from an enlarged photograph of a thin slice of granite. It shows the minerals crowded together irregularly. FIG. 226. Piece of white marble, a crystalline aggregate of a single mineral (calcite). The bright areas are due to reflections from the cleavage surfaces of individual grains. FIG. 227. Thin section of granite, illustrating a granular aggregate of several different minerals, viz: quartz (clear), soda-feldspar (striped), potash feldspar (cross-barred) and biotite (black). Magnified about 5 diameters. None of them possess crystal forms. Figure i (page 3) is a crystal group, composed of well-defined crystal individuals CRYSTAL AGGREGATES 145 of calcite, each bounded by its own planes except at its point of attachment. Classification of Crystalline Aggregates. The classifica- tion of crystalline aggregates is not in itself a matter of great importance. In discussing aggregates of crystal particles, however, it is convenient to make use of terms that will indicate briefly what would in ordinary language require a clause for its description. For descriptive purposes, then, we may classify aggregates with respect to the size of their constituent grains, with respect to their manner of development, and with respect to the strength of their cohesion. FIG. 228. Radiating groups of wavellite crystals on a rock surface. According to the size of the component grains, an aggregate may be phanero-crystalline when its particles are large enough to be seen by the naked eye, or crypto-crystalline when they may be detected only with the aid of a microscope. Phanero-crystalline aggregates may be coarse-grained, medium-grained, or fine- grained. According to the manner of development of its individual components, an aggregate may have a structure, which is described as granular, when the particles are about equally developed in all directions, as are the grains in a granite or a coarse limestone; lamellar, or platy, when the grains are scaly, platy, or tabular, 146 GEOMETRICAL CRYSTALLOGRAPHY as in the case of a mica-schist; or fibrous, when the components are all acicular, as in the case of asbestus ((MgCa)SiO 3 ). The components of a fibrous aggregate often tend to group FIG. 229. Globular mass of limonite fibers. themselves around certain centers from which they radiate, producing masses that imitate the shapes of common objects. When the fibers radiate from a point and are confined to a single FIG. 230. Botryoidal groups of limonite fibers. plane, like the spokes in a wagon wheel, they produce radial aggregates (Fig. 228). When the radiation is outward in all directions from a point, globular forms result (Fig. 229). These CRYSTAL AGGREGATES latter in turn may aggregate, forming a bunch of globules, when its form is known as botryoidal (Fig. 230). Often the radial grouping is incomplete or the fibers may not diverge regularly. In this case a sheaf-like bundle may result, as in the case of the FIG. 231. Sheaf-like group of stilbite crystals. mineral stilbite (calcium aluminium silicate with water) (Fig. 23 1 )- When the fibers radiate from a line they produce a stalactitic growth, or a stalactite (Figs. 232 and 233). FIG. 232. Stalactite of limonite. If the cohesion between the particles of an aggregate is strong, whatever may be the shape of the aggregate, it is said to be compact; when the cohesion is slight, the aggregate is friable. Parallel Growths. Frequently crystals are so grouped that 148 GEOMETRICAL CRYSTALLOGRAPHY one or all of the axes of the different individuals are parallel. The most complete parallelism occurs in the case of individuals of the same substance. Often two crystals of the same substance occur side by side in contact, with all the axes of the one parallel FIG. 233. Cross section of group of stalactites of limonite and goethite (hydroxides of iron.) FIG. 234. Group of barite crystals with axes parallel. (After Tschermak.} FiG. 235. Group of quartz crystals merging into single crystal below. (After Tschermak.] FIG. 236. A number of small crystals of quartz grouped to form a single large crystal. to the corresponding axes of the other, as in the case of barite (BaSO 4 ) (Fig. 234). Sometimes the contact surfaces are small, when the crystals just touch. At other times the crystals appear to penetrate one another, only a portion of each crystal being CRYSTAL AGGREGATES 149 visible. Figure 235 is a group of quartz crystals the lower portions of which are merged into a single crystal. In some cases many small individuals (sub-individuals) are grouped side by side, forming a large crystal (Fig. 236), or are arranged along a common axis in such a way that only the individual at the two ends of the group exhibit complete faces (zircon, ZrSiO 4 , Fig. 237), while the intermediate individuals appear as thin plates or lamellae crowded between these. When the lamellae of the intermediate individual are very thin they produce the effect of striations on the faces of a single individual. These striations are caused by the repetition of the interfacial edges on consecutive individuals a condition known as an oscillatory combination. Again many minerals form branching aggre- gates, the branches of which are composed of many small crystals arranged in parallel position (Fig. 238). The angles between the branches correspond in value to the angles between the crystallographic axes of the little individuals or to some other equally characteristic angles. These are often known as FIG con crystal com- posed of several individuals a r - ranged in a verti- cal pile. &x* FIG. 238. Skeleton crystals of argentite in section parallel to cubic face (a). Portion of same magnified (&). Showing grouping of tiny octahedrons arranged in lines parallel to the crystallographic axes. (After Tschermak.) dendritic growths. They occur in moss agate and in many glassy volcanic rocks. A druse is a crust composed of small crystals implanted side by side in approximately parallel positions upon some other mineral or upon a rock surface (Fig. 239). 150 GEOMETRICAL CRYSTALLOGRAPHY More frequent even than the parallel growth of similar mineral species is the case of parallel growths of crystals of different sub- stances, which are, however, analogous in composition. Sub- stances like calcite (CaCO 3 ) and magnesite (MgCO 3 ) possess similar though not identical crystal forms. Such substances are known as isomorphous substances (see page 229). These often crystallize together so that the axes of the crystals of both sub- stances are approximately parallel. Often the crystallization is so intimate that the different substances cannot be detected even under high powers of the microscope. Calcite, for instance, often FIG. 239 Coating, or druse, of small crystals of smithsonite (ZnCO 3 ) on massive smithsonite. contains small quantities of magnesium, which may occur in the compound as intermixed magnesite. At other times a nucleus of one substance may be surrounded by an envelope of another substance, and this in turn by an envelope of still a third substance. A cross section through such a growth will show a nucleus sur- rounded by concentric zones of varying composition. This is known as a zonal growth. It is well exhibited in the garnets, the feldspars (Fig. 240) and the pyroxenes, where the difference in composition of the different zones is indicated by a difference in color or by the effect of the different layers upon polarized light. A third class of partial parallelism is noted in the association of mineral species of entirely different compositions, possessing CRYSTAL AGGREGATES entirely different crystal forms. Little prismatic crystals of ruiile (TiO 2 ), for instance, are often implanted on tabular crystals of hematite (Fe 2 O 3 ) in such a way that the vertical axes of the rutile are parallel to one of the lateral axes of hematite (Fig. 241). FIG. 240. Section of zonal orthoclase in rock. Magnified 7^ diameters. (After Rosenbusch.) Many other instances of a like association of different mineral species are known, as, for instance, pyrite (isometric, FeS 2 ) on marcasite (orthorhombic FeS 2 ) (Fig. 242) and albite (NaAlSi 3 O 8 ) on orthoclase (KAlSi 3 O 8 ) (Fig. 243). FIG. 241. Small crystals of rutile attached to crystal of hematite. FIG. 242. Little crystals of pyrite implanted on marcasite. (After Linck.} Twinned Crystals. Often two or more crystals or parts of crystals are so grouped that they are symmetrical to each other with respect to a plane between them, that is not a plane of symmetry for either individual. These groups, or twinned crystals (Fig. 244), are of great importance crystallographically, 152 GEOMETRICAL CRYSTALLOGRAPHY as the nature of the twinning is very characteristic for different substances. Twinning Plane. The twinning plane is the plane about which the twinned individuals are symmetrical (plane ABCD in Fig. 244) . It can never be a plane of symmetry for the individuals, for in this case a parallel growth would result, but it may be any other crystallographic plane possible. Usually the twinning plane is one with very simple indices. The twinning axis is a line about which one of the twinned individuals or parts of individuals may be supposed to be revolved and brought into a parallel position with the other. The twin- ning axis is usually normal to the twinning plane. The plane of FIG. 243. Crystals of albite on orthoclase with vertical axes of both parallel. (After Linck.} FIG. 244. Contact twin with twinning plane and composi- tion faces the same, viz.: ABCD. union between the two twinned parts is the composition face, and this may or may not be coincident with "the twinning plane. In most cases of twinned crystals there is observed a re-entrant angle between certain, of the contiguous planes on opposite sides of the composition face, which angle serves as a distinguishing mark of twinning (see top of crystal in Fig. 244). Contact Twins. When parts of two crystal individuals are united in a plane in such a way that practically all of each indi- vidual is on one side only of the plane, the twin is known as a contact twin. In figure 244, which represents a contact twin of monoclinic CRYSTAL AGGREGATES 153 gypsum (CaSO 4 +2H 2 O), the twinning plane is parallel toooPoo, which is also the composition plane. In albite, which is triclinic (Fig. 245), the twinning plane is ooPoo(M), and the composition plane the same. In orthodase, which is monoclinic (Fig. 246), the twinning plane is oo P oo , and the composition face is oo P oo (M) (corresponding to ocPoo in albite). Penetration Twins. Frequently the twinning individuals penetrate each other, so that they cannot be said to have any composition face they are apparently grown through one an- other. Such twins are known as penetration twins. FIG. 245. FIG. 246. Twins of feldspars possessing different twinning planes but corresponding com- position faces. The twinning plane in Fig. 245 coincides with the composition face. In Fig. 246 the two are at right angles. Figure [247 illustrates an interpenetration twin of fluorite (CaFl 2 ) in which two cubes are so placed that they are sym- metrical about a plane parallel to an octahedral face. Figure 248 is an interpenetration twin of orthodase with the same law of twinning as in figure 246; i.e., ooPoo is the twinning plane and ocP oo the composition face. When the parts of such twins are of a low grade of symmetry and by their intergrowth tend to produce a body with a higher grade of symmetry, the resulting twin is known as a supplementary twin. Thus positive and negative hemihedrons by twinning may produce a form with the geometrical symmetry of a holohedron. Figure 249 represents the twinning of two right-handed tetartohedral crystals of quartz reproducing a form with hemihedral symmetry. GEOMETRICAL CRYSTALLOGRAPHY Repeated Twins. Although the term twin suggests that twinned crystals consist of two parts in the twinning relation with respect to one another, it nevertheless often happens that to the FIG. 247 . Interpenetra- tion twin consisting of two cubes of fluorite twinned about an octahedral plane. FIG. 248. Interpenetra- tion twin of feldspar with same twinning plane and composition face as in Fig. 246. A C B FIG. 249. Supplementary twin of quartz (C), produced by combination of two right-hand crystals (A and B) of which one with respect to the other is revolved about the c axis through an angle of 60. (After Grotti). two a third, a fourth, etc., are added, each possessing a twinned relation to those contiguous to it. Thus trillings, fourlings, etc., are produced. Repeated twinning may take place in either one^of two ways. CRYSTAL AGGREGATES The twinning planes between the contiguous individuals may be parallel to each other or they may not. In the former case the alternate individuals are in parallel posi- tion, as is indicated in figure 250, which represents a series of five orthorhombic aragonite (CaCO 3 ) plates, twinned about parallel planes of oo P. A cross section of such a twin is pictured in figure 251. Twins of this ^kind are known as polysynthetic twins. Each plate is called a lamella, and successive lamellae are in reversed positions with respect to their neighbors. Fig. 252 is a polysynthetic twin of albite twinned in the same way as is represented in Fig. 245 (page 153). FIG. 250. FIG. 251. Repeated twin of aragonite (Fig. 250) and cross section of same (Fig. 251) showing successive twinning planes parallel. Often a series of very thin twinned lamellae (Fig. 253) produces a striation on the faces of crystals or of cleavage pieces. This striation is well exhibited by cleavage pieces of triclinic feldspar (Fig.- 254). Light will be reflected in the same direction by alternate lamellae, so that, when the specimen is held in such a way as to catch the light from a distinct source, one set of lamellae will appear glistening while the intermediate lamellae are dull. Upon turning the specimen through a slight angle the reflecting lamellae become dull while those that were originally dull become bright. This phenomenon is the oscillatory twinning referred to on page 136. When the twinning planes between the successive individuals of a repeated twin are alternate planes belonging to the same form, the twin turns on itself, producing a circular or wheel-shaped group, known often as a cyclic twin. 156 GEOMETRICAL CRYSTALLOGRAPHY Figure 255 is an illustration of an aragoniie trilling in which the two symmetrical planes of ocP become successively the twinning plane. By further repetition of this twinning a nearly FIG. 252. Polysynthetic twin of albite with thin lamella in center. FIG. 253. Diagrammatic sketch of polysynthetic twin of albite with many lamellae. complete circle or hollow cylinder may be formed. Figure 256 is the cross section of such a repeated twin in which four individ- uals are twinned. The examples of repeated twinning given above are all FIG. 254. Twinning striations on cleavage surface of oligoclase. The bands are due to the alternation of lamellae that reflect light differently. Natural size. illustrations of contact twins. Interpenetration twins are also repeatedly twinned, the groups taking the form of a rayed star or of a bundle of symmetrical plates. Figure 257 is a repeated twin of cerussite (PbCO 3 ) an ortho- CRYSTAL AGGREGATES 157 rhombic mineral, in which the brachy-prism oo ?3 is the twinning plane; and figure 258, a twin of the orthorhombic chrysoberyl (BeAl 2 O 4 ), with the brachydome Poo the twinning plane. Figure 259 represents a very complicated interpenetration twin \ FIG. 255. Cyclic trilling of aragonite with successive twinning planes alternate oo P faces. Compare Fig. 250. FIG. 256. Cross section of arago- nite fourling with successive twinning planes alternate ooP faces. Com- pare Fig. 251. of monoclinic crystals of phillipsite ((CaK 2 )Al 2 Si 5 O 14 + sH 2 O). The three columnar portions of the group are twinned about the same plane (A- A). Each column is composed of what are apparently two individuals twinned about another plane (B-B), FIG. 257. Repeated in- terpenetration twin of cerussite. FIG. 258. Repeated in- terpenetration twin of chryso- beryl. but each of these seeming individuals is in reality a combination of two individuals twinned about a third plane. In the group there are, therefore, twelve individuals twinned according to three different laws. 158 GEOMETRICAL CRYSTALLOGRAPHY Mimicry. By the repeated twinning of crystals of a low grade of symmetry a group is often produced which appears to be a simple crystal of a higher grade of symmetry than that of its FIG. 259. Complicated inter- penetration twin 'of phillipsite. Twelve individuals twinned. See text for explanation. FIG. 260. Simple crystal of aragonite (orthorhombic) with coP(w), Pct> (&) and PS (A). components. Thus the orthorhombic mineral aragonite (CaCO 3 ) often crystallizes in prismatic crystals, bounded by the forms oo P (m), co Poo (b), and Poo (k) (Fig. 260). A cross section a = 116 16 FIG. 261. Cross section of trilling of aragonite with twin- ning plane alternate ooP faces. See also Fig. 256. FIG. 262. Trilling of aragon- ite producing a group that re- sembles an hexagonal prism terminated by basal planes. A reentrant angle is seen on the right. Natural size. through a single crystal has the outline shown in one of the individuals in figure 261, in which the angle between the ooP faces is 116 16'. Trillings formed of three of these crystals twinned CRYSTAL AGGREGATES I S9 parallel to ooP have a cross section like figure 261. When the spaces between the dotted lines in the figure and the body of the crystal become filled with mineral substance, the trilling strongly resembles an hexagonal prism. The resemblance is made striking by the close approximation of the angles and /? to the angles of the hexagonal prism (120). The angle a=n6 16' and /?=i27 28'. Figure 262 is the reproduction of a photo- graph of such a trilling. The interpenetration twin of chryso- beryl (Fig. 255) also simulates the hexagonal symmetry. CHAPTER XIV. AMORPHOUS SUBSTANCES AND PSEUDOMORPHS. Amorphous Substances. Although the great majority of chemical compounds possess definite forms, there are some to which such forms seem to be entirely lacking. Bodies of this kind possess neither the geometrical properties of crystals nor do they have the physical properties peculiar to crystallized bodies. Their internal structure has not the regularity of that of crystals. Such substances are said to be amorphous, or they are described as colloids. Their crystallizing power is so weak that it is not capable of causing the molecules in which it resides to group themselves in accordance with the symmetry of any crystal system, except, perhaps, under the most favorable conditions. Under ordinary conditions this power is not exerted sufficiently to effect any result, and so the material is put together according to no definite plan, and consequently it possesses no definite external form. The shapes exhibited by such substances are the result of external conditions or of forces not inherent in the molecules of the substance. They are largely, if not entirely, accidental. Pseudomorphs. Idiomorphic forms of crystals are deter- mined by the action of certain forces, called, for lack of a more definite name, crystallizing forces, which appear to be inherent in the substance of which the crystals are composed. The forms produced by their action are just as characteristic of the crystal- lized material as are its chemical reactions with other substances. Very frequently, however, substances are met with possessing definite crystal forms that are different from those which they usually possess, but which are similar to forms possessed by some other substance. The unusual forms originally belonging to some pre-existing substance and have been appropriated by the substance now possessing them. For instance, the mineral limonite (Fe 4 O 3 (OH) 6 ) is usually in globular or botryoidal forms (see Figs. 229 and 230). Sometimes, however, it occurs in cubes (Fig. 263), 1 60 AMORPHOUS SUBSTANCES AND PSEUDOMORPHS l6l which are known to be the forms in which pyrite (FeS 2 ) crystallizes. Bodies possessing forms borrowed from other substances are known as pseudomorphs (^ev&js, false, and p>oprj, form). This term is applied not only to the form itself, but as well to the substance exhibiting it. In the latter sense a pseudomorph is a body possessing the form of one substance and the chemical and physical properties of another. Pseudomorphism is the assumption by one substance of the form of some other pre-existing one. Explanation of Pseudomorphism. The explanation of pseudomorphism is comparatively easy. A substance possessing its own distinctive form may be changed by the action of the carbon dioxide, the oxygen, or the moisture of the atmosphere, or by some other agency into another substance differing from the original substance in nearly all of its morphological and physical properties. The material of the original substance is completely replaced by the new substance, but its external form remains unchanged. In such cases there results a pseudomorph. The mineral cuprite (CuO) often forms little octahedra. By exposure to the atmosphere cuprite changes to malachite (CuCO 3 . Cu(OH) 2 ), a monoclinic mineral crystallizing in acicular mono- clinic crystals. When the change from the cuprite to the mala- chite takes place slowly, the former mineral is replaced, molecule for molecule, by the latter, the result being a mass of malachite with the outward form of the cuprite, or a pseudomorph of malachite after cuprite. The malachite is in reality monoclinic, as may be learned by examining it optically, but it possesses the shape of a regular crystal. In the same way we find gypsum 162 GEOMETRICAL CRYSTALLOGRAPHY (CaSO 4 +2H 2 O) pseudomorphs after anhydrite (CaSO 4 ), limonite (Fe 4 O 3 (OH) 6 ) pseudomorphs after pyrite (FeS 2 ), etc. Petrified wood may likewise be described as a pseudomorph of opal (SiO 2 +Aq) after wood. Paramorphs. Sometimes what is apparently the same chemical compound occurs in two different forms in nature it is dimorphous, crystallizing in one form under certain conditions and in an entirely different form under other conditions. If, after the formation of crystals of such a substance, the conditions change, the entire mass of the crystal may pass over into the second form, producing a pseumodorph of the second substance after the first one. The external form of the second substance is now exactly similar to that of the first one, but its molecular structure is entirely different, thus giving rise to a genuine pseudomorph of a kind that has been distinguished by the name paramorph. A paramorph is a pseudomorph of one form of a dimorphous body after the other form. One of the most familiar illustrations of a dimorphous sub- stance is sulphur, which separates from a solution in carbon bi- sulphide as orthorhombic crystals, and from a molten mass as monoclinic needles. When allowed to stand at the normal tem- perature of the air, the monoclinic variety passes into the ortho- rhombic variety. The material still possesses the acicular form of the monoclinic sulphur, but its molecular structure is that of the orthorhombic variety. Here we have an example of a paramorph of orthorhombic sulphur after the monoclinic variety. Two Classes of Pseudomorphs. The processes described above as originating pseudomorphs are chemical, hence pseudo- morphs produced by them are known as chemical pseudomorphs. There is another class of pseudomorphs, however, known as mechanical pseudomorphs. The forms of these bodies are not produced by the replacement of the substance of a crystal, par- ticle by particle, by the pseudomorphing substance. They are produced simply by the filling of a mould left by the solution of some pre-existing crystal. The original crystal may become incrusted with some insoluble material. Its substance may then be dissolved, leaving a cavity of the shape of the crystal. If this AMORPHOUS SUBSTANCES AND PSEUDOMORPHS i6 3 cavity be filled with a new substance, and then the enveloping material be removed, the new substance will necessarily possess the form of the original crystal. At Girgenti, in Sicily, pseudomorphs of calcite (CaCO 3 ) after sulphur are sometimes met with. The origin is explained as follows: The sulphur crystals were incrusted with a coating of barite (BaSOJ. The temperature in the neighborhood rose until the sulphur melted and disappeared, leaving a mould of itself constructed of barite. By the infiltration of a solution contain- ing calcium carbonate and the deposition of this substance as cal- FIG. 264. Fossils. Pseudomorphs of dolomite after Mollusks and Coral. cite in the mould, the cavity was filled. Upon the removal of the barite coating a mass of calcite was left with the form of the sul* phur crystals. Fossilization. Fossils are pseudomorphs of mineral sub- stance after organisms or parts of organism. The processes of fossilization are exactly analogous to those of pseudomorphism. The original organism may have been replaced, molecule by molecule, with the fossilizing substance, or it may have been dis- solved from the rock in which it was imbedded, leaving a cavity of its own shape, which afterward was filled with mineral substance. Fossils produced by the latter process preserve only the external form of the original organism, while those produced by replacement often retain even the minute internal structure of the original. The fossilizing substance is usually calcite (CaCO 3 ), dolomite ((CaMg)CO 3 ), or silica (SiO 2 ) in some form, though fossils composed of other minerals are not uncommon (Fig. 264). CHAPTER XV. CRYSTAL PROJECTION. Projection. By the term projection in crystallography is meant the representation of crystals on a plane surface; i.e., on the surface that contains their figures. There are several graphic methods by which the planes on crystals are represented, among which are the linear projection and the spherical pro- jection. These exhibit the relations of the planes to one another without reference to the shape of the crystal bounded by them. Other methods of projection one of which is the clinographic projection represent the crystal as it appears to the eye under certain conditions. This projection is a picture. The Linear Projection. This method of projection repre- sents each face on a crystal by its line of intersection with the plane on which the projection is made, when that face is assumed to pass through the unity of the vertical axis. The plane taken as the plane of projection is usually that which includes the lateral axes of the crystal. The projection then appears as a number of straight lines that run in different directions across this plane and at different distances from the point representing the center of the crystal. Figure 265 shows the projection of the icositetrahedron. The planes are all assumed to pass through the unity on the vertical axis because the intercepts on this axis cannot be indicated directly in the projection. They must therefore be indicated indirectly and this is done by imagining them moved parallel until they pass through the unity on c, and drawing their inter- sections with the plane of projection by joining the intercepts which they make on the lateral axes in this position. The planes thus projected are parallel to the planes on the crystal. The ratios of their intercepts are the same as they were before the imagined movement, and consequently, from the crystallographic 164 CRYSTAL PROJECTION 165 point of view, they are the same planes. Practically, the method consists in writing the intercepts of the plane to be projected in the form of a ratio and reducing to unity the term relating to the c axis. By laying off on the lines representing the lateral axes distances corresponding to the reduced intercepts on these axes and connecting them by a straight line, the projection of the plane is obtained. In all the systems with three axes the lateral axes are repre- sented in the projection by two lines perpendicular to each other. They are usually drawn dotted to distinguish them from FIG. 265. Form 2O2 and its linear projection. the projections of the planes which are drawn solid. In the hexagonal system the axes are represented by three lines inter- secting at 60. In the systems with equal unities on the lateral axes equal distances are laid off on the projection of these axes to represent unity. In the other systems the distances that must be laid off to represent the unities must correspond to the ratio between the unities on a and b for the crystal to be projected. Usually only the upper half of the crystal is projected, as this projection in most cases represents the lower half as well. In the case of certain hemihedral and tetartohedral forms, in which the planes on the two halves are differently related to the axes, two projections must be made to represent the entire form. These, however, may be indicated in the same figure by designating the 1 66 GEOMETRICAL CRYSTALLOGRAPHY upper and lower planes by some conventional sign. For instance, the projections of the upper planes may be drawn solid and those of the lower planes may be dotted. Suppose the form 262 is to be projected. The lateral axes are first indicated as two dotted lines perpendicular to one another (see Fig. 265). Convenient equal lengths are laid off as unities. The form consists of 12 planes above the plane of projection, three of which are in each octant. In the octant in which all the intercepts are positive the symbols of the planes are : 20, : b : 2C-, a : 2b : 2c\ 2a : 2b : c, or a : i/2b : c\ 1/20, : b : c; 20, : 2b : c, when the intercept on c is reduced to unity. For the projection of the first plane lay off on the right-left axis a distance equal to 1/2 the length decided upon as unity and connect this point by a straight line with the unity distance on FIG. 266. Linear projection of dodecahedron. the front-back axis (A A). For the projection of the second plane lay off 1/2 the unity distance on the a axis and connect this by a straight line with the unity point on b (B B). For the projection of the third plane lay off two unities on a and b and connect by a straight line (C C). By a similar process the projection of the planes in the other three upper octants are drawn and the figure is completed (see Fig. 265). Planes that are parallel to the vertical axis must necessarily pass through the intersection of the axes in the projection, since, CRYSTAL PROJECTION i6 7 when such planes are made to pass through the unity on c, they cut this axis throughout its entire length. Thus the pro- jection of oo O consists of two lines passing through the unity distances on a and parallel to b, two lines passing through the NJ P == oP x k = Co/' 55 c M= GO/'C T =- o = P w = 2P c). FIG. 267. Crystal of orthoclase with linear and spherical projections. unities on b and parallel to a, and two lines passing through the center of the projection and bisecting the angles between the axes. The symbol of the first two lines is a : oo b : c; of the second pair, GO a : b : c; and of the third pair a : b : ooc. (See Fig. 266.) 1 68 GEOMETRICAL CRYSTALLOGRAPHY Spherical Projection. In the spherical projection the position of each plane on the upper half of the crystal is repre- sented by the point of intersection of a perpendicular to the center of its face with the surface of a hemisphere at whose center the crystal is supposed to be. The hemisphere is then projected on a plane passing through its equator. The projection appears as a lot of dots arranged with the same symmetry within a circle as that of the planes on the crystal. Figure 267 shows the projection of a crystal of orthoclase (a monoclinic mineral) by the linear and the spherical methods. Crystal Drawing. The representation of crystals as they appear in nature (clinographic projection), or the drawing of crystal figures such as are used in this book, is different from ordinary perspective drawing in that lines which are parallel on the crystal are made parallel in the representation. FIG. 268. Illustration of method of drawing the dodecahedron. In constructing a crystal drawing the axes are first represented as they would appear if the eye were viewing them from the right and from above. A linear projection is then made on the lateral axes, and lines representing the directions of the interfacial edges between contiguous planes are drawn from the ends of the vertical axes to the points in the projection where the lines representing the planes intersect. Figure 268 shows the projection of the axes in the regular sys- tem, with the projection of the planes of the dodecahedron ( O) upon the plane of the lateral axes, and the completed drawing made from this projection. CRYSTAL PROJECTION 169 Projection of the Crystal Axes. The most important step in the drawing of any crystal is the projection of its axes, because these serve as the foundation upon which the drawing is constructed. After the preparation of the per- spective view of the axes the completion of the drawing is comparatively easy. Different methods of making the projection of the axes are em- ployed, the choice between them depending upon those features of the crystal that it is desired to emphasize. For general pur- poses the vertical axis is repre- sented as a vertical line, and the lateral axes are drawn in perspec- tive on the assumption that they are viewed by the eye at a certain angular distance (8) to the right of the center of the crystal and elevated a certain angle (e) above it. Different values may be assigned to 8 and , and as a result different views of the crystal may be shown. It is usual, however, to assume such values as may be expressed by a simple ratio between equal axes after projection. If the ratio between the two axes OI and OK' (Fig. 270) is i : 3 and the ratio between AI and OI is i : 2, then 8= 18 26' and ^ = 9 28'. Figure 269 represents the normal position of the lateral axes in the isometric or tetragonal system as viewed from the top of the crystal; i.e., along the direction of the vertical axis, c. If viewed from the direction of the axis AA(a) this will appear as a point, while BB(6) will appear in its true length. After revolution to the left through the angle AOA' ( = 8= 1 8 26') the axis AA will have the position A' A'; i.e., its front half will be lengthened to OI, and BB will assume the position B'B'; i.e., its right half will be shortened to OH. If, now, the eye be elevated (at the angle ?) the lines A'l, AO, and B'H will be projected below I, O, and H to distances propor- tional to the lengths of the respective lines. If 8=18 26' and i yo GEOMETRICAL CRYSTALLOGRAPHY = 9 28', then OI : OH=i : 3 and a'l : OI as i : 2 (in which a'l is the projection of AI below I). Projection of the Axes for the Isometric System. If the values for 8 and e be assumed as above, then the method of con- struction of the isometric axes is as follows: Draw two lines LL' and KK' at right angles to one another (Fig. 270). Make KO = K'O = unity on b, and divide KK' into three equal parts. Draw verticals through the four points thus obtained on KK', and below K' lay off K'H = 1/2 K'O. Draw HO, which will give the direction of the front lateral axis. Its length will be that portion of this line included between the two inner verticals, A and A'. Draw AS parallel to K'O and connect the points S and O. From the intersection of this line with the inner vertical, T, draw TB parallel to K'K. From point, B, thus obtained draw the line BB' through O. This will be the second lateral axis, a. Below K, lay off KQ=i/3 OK and make OC = OC' = OQ; then CC' will be the length of the vertical axis.* Projection of the Axes for the Tetragonal and Ortho- rhombic Systems. The axes constructed for the isometric sys- tem may be readily adapted to both the other systems with rect- angular axes by merely laying off portions of the lines AA' and * In order to avoid the necessity of projecting the axes each time a drawing is to be made, it is advisable to construct a set on a piece of cardboard and prick holes at the ends of the axes and their point of intersection. The axes can then be trans- ferred to drawing-paper by making dots through these holes and connecting them by straight lines. The relative lengths of equal unity distances will be indicated by the positions of the dots. Longer or shorter unity distances are secured by increas- ing or diminishing proportionately the lengths thus obtained. It is important that the construction be made with all possible accuracy, otherwise the completed figures may be distorted. This is secured in part by making the original drawing of such a size (e.g., by making the entire length of the b axis = 4 inches) that small errors will be practically eliminated when the lengths of the axes are reduced to the dimensions ordinarily employed in drawing crystals. FIG. 270. Construction of isometric axes. CRYSTAL PROJECTION 171 CC', which are proportional to the lengths expressed in the axial ratios of the crystals to be figured. For instance, if the axial ratio of the crystal to be drawn is a : b : c=i.$ : i : 2.4, proceed as follows : Transfer the perma- nent projection of the axes to the paper upon which the drawing is to be made. Take proportional lengths of the axes as thus constructed if more convenient than the entire lengths. These distances will represent ratios of i : i : i on the three axes. In- crease the length on a by . 5 of itself and that on c by i . 4 of itself. The resulting lengths will have the rela- tions i . 5 : i : 2 . 4, or the axial ratio desired. Indicate these by dots and treat them as the unity inter- sections on the several axes. In the case of a tetragonal crystal like zircon, the axial ratio of which is a : c : : i : .64, the two lateral axes remain unchanged, while the vertical axis must be made .64 of the length CC'. For an orthorhombic crystal the axis BB' alone remains un- changed, while AA' and CC' are both changed to the propor- tionate lengths belonging to the substance in question. Projection of theMonoclinic Axes. To project the inclina- tion, /?, of the clinoaxis, construct the axes as in the isometric system, and then lay off Oc=OC.cos /?, and on OA' lay off Oa = OA'. sin /5 (Fig. 271). From c draw a line parallel to OA', and from a another parallel to OC. From their intersection, a line (DD') drawn through O will give the direction of the clino- axis. The directions of the other two axes remain unchanged. The relative lengths of the axes must now be laid off, accord- ing to the axial ratio of the substance, as in the orthorhombic system. Projection of the Triclinic Axes. In this case all three axes of reference intersect obliquely b Ac= <*, a Ac = fi, a A 6 = 7-. If we start with the isometric axes, the first step in their adaptation FIG. 271. Construction of mono- clinic axes. 172 GEOMETRICAL CRYSTALLOGRAPHY D' FIG. 272. Construction of triclinic axes. to the triclinic system is to obtain the direction of the two vertica axial planes or pinacoids. To do this, lay off (Fig. 272) on OB Ob = OB. sin (/> ( being the angle ooPoc A ooP ex, which is evidently not the same as ?-), and on OA, Oa=OA. cos . The line drawn from the angle d of the parallelogram adbO through O will give the direction of the macropinacoidal section, DD'. To obtain the direction of the macroaxis (b), lay off on OD'> Od' = OD. sin a; and on OC, Oc=OC. cos . From the parallelogram, d'OcK', thus obtained, the diagonal, K'K, gives the macroaxis. In a similar manner, the brachyaxis (a), HH', is found by laying off on OA', Oa' = OA. sin /?: and upon OC, Oc' = OC. cos /?. The vertical axis CC' and the lateral axes HH' and KK' thus obtained are the axes of a triclinic crystal in which a : b : c = i : i : i. Their relative lengths must now be given them in accordance with the axial ratio of the sub- stance, just as in the ortho- rhombic and monoclinic systems. Projection of the Hexagonak Axes. Con- struct an orthorhombic set of axes whose axial ratio, a : b : c, is>/3 (=1.732) : i : c (c being given the value of the vertical axis belonging to the substance to be drawn) (Fig. 273); connect the extremities of the two lateral axes, and, in the rhomb thus formed, the obtuse angles, at the ends 4-a, FIG. 273. Construction of hexagonal axes. CRYSTAL PROJECTION 173 of the b axis, will be exactly 120. If .lines be now drawn parallel to b, through points on the axis a, half way between its extremities and the center o, the rhomb will be converted into a hexagon, with all of its angles exactly 120. If we connect the diagonally opposite angles of this hexagon, we shall obtain the projection of the hexagonal axes required. Construction of the Drawing. Having made a projec- tion of the axes the next step is to transfer to the lateral axes the plane projection of the faces on the crystal to be drawn (see Fig. 268). Remembering that all the lines in this projection repre- sent the traces of the planes when they are supposed to pass through the unity point on c, it follows that this point is common to all the planes represented in the projection. The outline of the crystal is shown by drawing the interfacial edges between neighboring planes. These edges are represented as lines in the drawing. It is only necessary ,then, to find one point other than the unity on c which is common to the two planes whose intersection is desired. This is the point of intersection of the two lines representing the planes in the projection. Thus we obtain two points which are at the same time in the two intersect- ing planes. The line joining them gives the direction of their interfacial edge. If, therefore, we draw a line from the point representing the intersection of the two planes in the projection to the unity point on c we have the direction of the interfacial edge of these planes. In cases when the intersecting planes are repre- sented by parallel lines in the projection, their interfacial edge is indicated in the drawing by a line parallel to these lines in the projection. It will be noted that the lines drawn by this procedure indicate only the directions of the desired intersections. They all diverge from the unity point on the vertical axes. In the crystal the inter- sections are not so distributed. They are lines joining certain points on the crystal surface which may or may not be at the unity point on the c axis. In constructing the drawing the lines with their proper directions must be drawn from points that have the relative positions of the corresponding points on the crystal. Usually a prominent face is first outlined by drawing the 74 GEOMETRICAL CRYSTALLOGRAPHY proper lines. Then from points on the outline of the face other lines are drawn to represent the interfacial edges that extend from it, and so on until the complete figure is produced. The size of the figure will be determined by the size of the first face outlined. Its proportion will in some cases be fixed by the symmetrical develop- ment of the figure. In most cases, however, the proportions must be controlled by noting the proportional lengths of the different interfacial edges on the crystal and making them correspond in the figure. In other words, the direction of the lines representing the interfacial edges are fixed by the projection, but the habits of the crystals represented are indicated by the relative sizes of the planes in the drawing or the relative lengths of the lines representing their interfacial edges. PART II. PHYSICAL CRYSTALLOGRAPHY. CHAPTER XVI. INTRODUCTION: PHYSICAL SYMMETRY AND PHYSICAL AGENCIES. Physical Symmetry. The material of which crystals are composed, like all other matter existing throughout the universe, is made known to our senses through its properties. The sub- stance of crystals obeys the same laws of physics that govern all other matter, but these laws manifest themselves a little differently than they do in non-crystallized bodies, because of the fact that the crystal particles are put together in a definite manner. In amorphous bodies physical forces produce exactly similar effects when acting in different directions, provided the bodies are homogeneous throughout. In crystalline bodies the case is different. In these the effects of forces vary with directions, and the variation is in the closest accord with the symmetry of the substance acted upon; i.e., with the arrangement of its component particles as expressed in the symmetry of its idiomorphic forms. Every geometrical plane of symmetry is also a physical one, and all geometrically equivalent directions are also equivalent with respect to physical agents. When any given force acts upon a crystal it will produce similar effects along parallel directions, and effects varying in degree along different directions which are not symmetrically disposed about planes and axes of symmetry. Of course a force like the force of gravity may act upon a crystal as a whole, and then the effect of symmetry will not be apparent. In all cases, however, where the force acts successively upon different particles of the crystal or when it is transmitted by these particles, the effect of symmetry is clearly noticeable. When considering the effects of forces acting upon the surfaces of crystals it must be kept in mind that the symmetry of the planes bounding a crystal is something quite different from the symmetry of the crystal itself. Only those crystallographic planes of 177 i 7 8 PHYSICAL CRYSTALLOGRAPHY symmetry that are perpendicular to the bounding planes can be planes of symmetry for these surfaces. It is impossible to con- ceive of a plane surface symmetrically disposed about a plane inclined to it. As an illustration of the relation existing between the sym- metry of planes and that of the crystals on which they occur, let us consider the basal planes on holohedral forms belonging to the different crystallographic systems (see Fig. 274). The top plane of the cube in the isometric system corresponds to the basal pinacoid in the other systems. In the isometric system this plane is sym- metrical with respect to the four planes inclined to one another at angles of 45 and intersecting in a common line, which is the FiG. 274. Diagrams illustrating symmetry of planes at terminations of the vertical axes in holohedrons of the different systems. vertical axis of the cube (a) . In the hexagonal system the basal plane is symmetrical with respect to six similarly intersecting planes inclined to one another at angles of 60 (b). The cor- responding face in the tetragonal system is similar to the cubic face in its symmetry. In the orthorhombic system it is sym- metrical with respect to two planes of symmetry perpendicular to each other (c). In the monoclinic system the basal plane is traversed by a single plane of symmetry (d), and in the triclinic system it is crossed by no symmetry plane (e). Forces acting upon the faces of a crystal will be governed in their effects by the symmetry of the faces. .From the symmetry of the effects produced the symmetry of the faces is disclosed and from the symmetry of several faces the 'symmetry of the whole form to which they belong may be deduced. Consequently, the grade of symmetry possessed by a substance may be dis- covered even when complete crystals are not obtainable. Physical Agencies. The physical forces producing effects PHYSICAL SYMMETRY AND PHYSICAL AGENCIES 179 that are of the greatest importance in determining the grade of symmetry of crystals may be classified as mechanical, optical, thermal, and electrical. The nature of these forces is a subject dealt with in physics. At present we are more directly concerned with the character and distribution of the effects they produce. These effects, as has already been stated, accord in their general character with the symmetry of the crystal upon which they act. The relations of the substance toward the forces are known as their properties. Thus we speak of the optical properties of crystals when we refer to their action with reference to the forces that produce light, of their thermal properties when we refer to their action under the influence of thermal forces, etc. CHAPTER XVII. MECHANICAL PROPERTIES OF CRYSTALS. Mechanical Properties. The mechanical properties of crystals are those they exhibit with respect to mechanical forces. They are discussed under the following heads: Elasticity, tenacity, cohesion, cleavage, fracture, hardness, and density. Elasticity. By elasticity is meant that property which causes bodies to resist forces tending to change their form. This power of resistance varies in different substances and varies along different directions in the same body, provided it is not amorphous. It is expressed by the coefficient of elasticity, which is the relation existing between the length of a standard-sized bar of the substance and the elongation it suffers under the influence of a given pull. For all amorphous bodies the coefficient of elasticity is equal in all directions; that is, bars of equal size cut from amorphous bodies in any direction will be equally elongated when subjected to the same strain. In crystals the value of the coefficient of elasticity varies according to the direction in which the stress is applied. In isometric crystals it is equal in directions parallel to the three crystallographic axes, and varies from this in other directions always being equal, how- ever, in parallel directions, and in directions that are symmetrical with respect to one another. The coefficient of elasticity in salt (NaCl), for instance, is as i to .7 in rods cut perpendicular to the cubic face and those cut perpendicular to the octahedral face. Although the symmetry of a crystallized body may be deter- mined by a study of its elastic properties even when crystals of it are not obtainable, nevertheless, since there are much more con- venient methods than this that may be employed for the purpose, the elastic properties are not made use of to any great extent. Tenacity. If the elasticity of a body is defined as the quan- 180 MECHANICAL PROPERTIES OF CRYSTALS l8l tity of resistance it opposes to deforming influences, we may de- fine tenacity as the quality of this resistance. With respect to tenacity, substances are distinguished as brittle, sectile, malleable, flexible, and elastic. A brittle substance is one that breaks into powder when cut with a knife, as does calcite (CaCO 3 ). A sectile substance may be cut, but it pulverizes under blows, as, for instance, gypsum (CaSO 4 +2H 2 O). A malleable substance flattens when hammered upon, as copper and other metals. A flexible substance will bend when subjected to forces properly applied, and will remain bent when the action of the FIG. 275. Cleavage piece of calcite showing cleavage cracks. forces ceases, as talc (H 2 Mg 3 (SiO 3 ) 4 ) and asbestus ((CaMg)SiO 3 ). An elastic substance will fly back into its original position when the force that bends it is removed, as mica (H 2 (KNa)Al 3 (SiO 4 ) 3 ). Cohesion. Cohesion is the resistance offered by bodies to the separation of their particles. Those special characters that are dependent upon the strength of this resistance are cleavage, fracture, and hardness. Cleavage. Many crystals possess a marked tendency to split along certain directions in preference to others, in consequence of differences in cohesive power in different directions. The planes along which such splitting occurs are known as cleavage planes (Fig. 275). They must be perpendicular to the direction of mini- mum cohesion and their perfection must depend upon the dif- 182 PHYSICAL CRYSTALLOGRAPHY ference in the cohesive force along different directions the greater the cohesion difference the better the cleavage. The cleavage planes are always parallel to planes that are crystallographically possible; that is, they are planes with rational indices. Moreover, if one crystal face is parallelized by a cleavage plane, all other faces belonging to the same crystal form are also parallelized by cleavage planes, and along all of these planes the cleavage is equally easy. This must be so because cohesion is a property of the molecules and in crystals the molecules are regu- larly arranged. Their geometrical forms and physical properties are but different expressions of this arrangement. When two different cleavages are present, i.e., when a crystal cleaves parallel to faces belonging to different crystal forms, the ease with which the cleavages are produced is unequal, and the character of the surfaces produced differs. In galena (PbS), for instance, cleavage is equally easy in three directions perpendicular to one another; i.e., in directions parallel to cubic faces the cleavage is cubical. In sphalerite (isometric ZnS) the cleavage is equal along planes parallel to dodecahedral faces it is dodecahedral. In barite (orthorhombic BaSO 4 ), on the other hand, there are two unequal cleavages. The most easy one is- parallel to oP and the most difficult one parallel to P. "The differences in the character of cleavages produced in a substance often serves to determine its system of crystallization. Anhydrite (CaSOJ, for instance, cleaves along three planes per- pendicular to one another, but with different degrees of perfec- tion. The differences in ease with which the cleavage takes place along the different planes serves to fix the symmetry of the sub- stance as orthorhombic. According to the ease with which cleavage is effected and the evenness of the surfaces produced by the cleavage, this is said to be very perfect, perfect, distinct, indistinct, imperfect, interrupted, or difficult. The relative ease with which cleavage is produced along dif- ferent planes is measured by cutting rods of a substance with their long axes parallel to different crystallographic lines, and then r training them until they break. The symmetry of the cleavage MECHANICAL PROPERTIES OF CRYSTALS 183 may be determined by measuring the force of the strain under which the cleavage is effected in a number of rods. Gliding. Many crystals possess a property which is analo- gous to shearing, but which differs from this in the fact that it occurs only along certain planes, which are perpendicular to the direction of maximum cohesion and which are known as glid- ing planes. Small portions of the crystals may be moved along the gliding planes without being separated from the unmoved FIG. 276. Diagrams illustrating shearing of ice. (After Linck.) A. Rod cut from ice crystal parallel to its vertical axis (cc). B. The same rod after being bent by loading its center. The vertical axis in the bent portion remains parallel to its original position. C. Explanation of method of bending. Thin slices of the ice shear downward. parts. In many instances the movement consists of a slipping along a series of parallel planes and is unlimited in amount, thus resembling true shearing. In other cases the movement is caused by a rotation of the molecules in a series of rows parallel to the gliding plane. Each molecule moves but a slight amount with reference to neighboring molecules, and the limit of movement is reached when the part of the crystal that has been deformed occu- pies the twinned relation to the undeformed part with the gliding plane as the twinning plane. The two types of gliding are well illustrated by ice and calcite (Fig. 276). If a bar cut from a crystal of ice parallel to its vertical 184 PHYSICAL CRYSTALLOGRAPHY axis be placed in a horizontal position and supported at its ends while weighted at its center, the middle portion of the bar will slowly sag downward. Examination of the bent portion will show that the sagging is not due to bending, as the crystallographic axis will be found to have suffered no deformation in the sagged portion of the bar. It will still remain parallel to its original position. The sagging must therefore be due to downward slipping of a large number of parallel lamellae (see Fig. 276). If, on the other hand, a sharp knife- __ blade is placed perpendicular against FIG. 277. Artificial twin of the blunt edge of a cleavage rhomb te produce, substance, a slice of the material will move without fracturing until it assumes a twinned position with reference to the rest of the calcite (see Fig. 277). The corresponding axes in the moved portion will no longer be parallel to their original positions, but they will be rotated into twinned positions. The plane between that portion of the calcite that has been moved and that portion which has not been disturbed is the gliding plane. o FIG. 278. Diagrams illustrating process of gliding in calcite. The circle drawn near the upper corner (A) is elongated into an ellipse after gliding (5). It is also the twinning plane. The fact that the horizontal rows of molecules have rotated is shown by the change of circles into ellipses during the movement (see Fig. 278). Secondary Twinning. Very frequently gliding is produced in opposite directions along a succession of parallel planes so that there results a series of lamellae each in the twinning position MECHANICAL PROPERTIES OF CRYSTALS 185 with respect to its neighbors. Polysynthetic twinning produced in this way is known as secondary twinning because brought about after the crystals exhibiting it were formed. It is a common phenomenon in the calcite grains composing marble (see Fig. 279). Percussion Figures. When a hard point is placed against the face of a crystal and then is tapped with a sharp stroke, cracks may be produced forming a star-like figure, the shape of which FIG. 279. Thin section of marble viewed in polarized light. The dark bars are secondary twinning lamella? due to gliding under the influence of pressure. Mag- nified about 5 diameters. is characteristic for many substances. The cracks are partings along definite crystal planes, and are closely related to the gliding planes. The percussion figure on the cubic face of halite (NaCl) is a four-rayed star with the rays parallel to the diagonals of the cubic face. The rays are cracks that are perpendicular to the face and therefore parallel to the four dodecahedral planes that truncate the vertical edges of the cube. On the octahedral face the percussion figure is three-rayed. A six -rayed star is produced in a similar manner on cleavage pieces of mica (Fig. 280) which crystallizes in the monoclinic system. The ray which is parallel to the plane of symmetry the clinopinacoid is larger than the others, and is called the i86 PHYSICAL CRYSTALLOGRAPHY characteristic ray. By its means the position of the clinopinacoid may be determined in plates of mica that show no crystal planes. Pressure Figures. If a very thin cleavage plate of mica is placed on a yielding support and pressed by a blunt point, another figure is produced which is also six-rayed when perfect (Fig. 281). Usually, however, some of the rays are missing and the figures consist of three or four cracks only. The rays of the pressure figure are perpendicular to those of the percussion figure. Cracks having the directions of the rays of the pressure figure are often observed in pieces of mica and sometimes triangular FIG. 280. Percussion figure on basal plane of muscovite. The mineral is monoclinic and the long ray is parallel to the plane of symmetry, ocPoo . FIG. 281. Pressure figure in basal plane of muscovite. The rays bisect the angles between those of the percussion figure. fragments of the mineral are found in which actual separation has occurred along these directions (Fig. 282). The phenomena are due to pressure exerted on the crystal substance while in the rocks. Parting. Regular breaking along planes which are not cleavage planes is known as parting. It differs from cleavage in that it occurs only in certain places; i.e., along the cracks produced by pressure, along the planes separating twinned lamellae, etc., while cleavage may take place equally well anywhere parallel to the cleavage plane (see Fig. 282). Fracture. When a force breaks a crystal in a direction which is neither a cleavage nor a gliding plane, or produces a break in an amorphous body, the separation takes place in an irregular way. MECHANICAL PROPERTIES OF CRYSTALS i8 7 This kind of breaking is known as fracture. It is described according to the character of the surfaces produced, as even, splintery, earthy, hackly, or conchoidal. A hackly fracture leaves surfaces that are ragged and rough, such as is exhibited by a broken piece of malleable metal. A conchoidal fracture leaves surfaces marked by concentric, or nearly concentric, curved lines, like the lines on many shells of molluscs (see Fig. 283). This kind of frac- ture is best exhibited by glass. The names applied to the other kinds of fracture are self-descriptive. Hardness. The hardness of a substance may be measured in a number of different ways, but the results obtained are not comparable. FlG 28 2. -Fragment of muscovite parted along the planes of the pres- Consequently, a satisfactory defini- sure figure. (After Linck.} tion of hardness is not yet possible. A harder substance will scratch a softer one. The miner- FIG. 283. Conchoidal fracture in obsidian. alogist Mohs proposed the names of the following ten minerals to serve as a scale to which to refer all other minerals with respect to hardness. The scale begins with a soft mineral and ends with the hardest substance known. 1 88 PHYSICAL CRYSTALLOGRAPHY MOHS'S SCALE OF HARDNESS? i-Talc (H 2 Mg 3 (Si0 3 ) 4 ) 2 Gypsum (CaSO 4 +2H 2 O) 3Calcite (CaCO 3 ) 4 Fluorite (CaF 2 ) 6 Feldspar (KAlSi 3 O 8 ) 7 Quartz (SiO 2 ) 8 Topaz (Al 2 F 2 SiO 5 ) 9 Corundum (A1 2 O 3 ) 5 Apatite (Ca 5 (PO 4 ) 3 F) 10 Diamond (C). A mineral that neither scratches any given mineral in the scale of hardness nor is scratched by it is said to have the same hardness as this; if it scratches one of the scale minerals and is scratched by the next harder one, its hardness is between that of the former and that of the latter. For example, a mineral that neither scratches quartz nor is scratched by it has a hardness of 7. One that scratches feldspar and is scratched by quartz possesses a degree of hardness between 6 and 7. Minerals with a hardness of 2 or under can be stratched with the finger-nail. Those whose hardness is between 2 and 4 can be scratched easily with the point of a knife. Those with a hardness of 4 to 5 cannot be scratched with a knife, but can easily be scratched with a good file. Only those minerals whose hardness is greater than 5 will scratch window-glass. Only those with a hardness above 7 will strike fire with steel. The degrees in the Mohs's scale are entirely arbitrary and the intervals between them are very unequal. The following table records the relative hardness of the minerals comprising the Mohs scale as determined by different men using different methods: RELATIVE HARDNESS OF CERTAIN MINERALS AS DETERMINED BY DIFFERENT INVESTIGATORS. Mohs 1 2 3 4 5 6 7 8 9 10 Franz !3-5 54 '35 390 670 5 4 o 1000 Pfaff I .IT. 12 .03 1C . -2 77 . T. Cl . C IQI 2<4 4 en IOOO Rosiwal 33 1-25 4-5 s 6.5 37 120 T 75 TOOO 140,000 Jaggar .04 .26 75 1.23 25 40 !5 2 IOOO MECHANICAL PROPERTIES OF CRYSTALS 189 Usual Method of Determining Hardness. The relative hardness of different substances is usually determined by measur- ing the force required to scratch their smooth surfaces. This is done by means of an instrument known as a ' skier ometer. One form consists of a sharp point of steel or diamond capable of being weighted and a movable platform that can be drawn beneath it. A plate of the substance to be tested is placed on the FIG. 284. Sklerometer an instrument for determining differences in hardness in different directions on crystal faces. platform and the point pressed down upon it. The force neces- sary to drive the point sufficiently deep into the substance to leave on it a scratch when the platform is moved is a measure of its hardness. One form of the sklerometer is shown in the accompanying figure (Fig. 284). Its essential features are the little car or movable platform (k), the point (m), the platform for the load (p), and the basket (g), containing a constant weight by which the car is drawn beneath the point. Differences in Hardness. As the result of experimentation with the sklerometer it is learned that not only do different sub- stances possess different degrees of hardness, but also that the 1 90 PHYSICAL CRYSTALLOGRAPHY same substance when crystallized offers greater opposition to a scratching agent acting in a certain direction than to the same agent acting along some other direction. The triclinic mineral cyanite (Al 2 SiO 5 ), for instance, resists scratching parallel to c much less effectually than scratching at right angles to this direction. Cyanite is harder in directions perpendicular to c than in directions parallel to this axis. In general it is found that crystals are equally hard in directions that are symmetrical with respect to each other, and that the hardness differs in directions that are not symmetrical. The differences in hardness in any crystal seem to be governed largely by the cleavage, scratches always being produced more easily parallel to the cleavage than across \\\\\\\\\\\ it. Since the cleavage is in accord with \\\\\\\\\\\ the symmetry of the crystal, its hardness must likewise be symmetrical. More- over, if the cleavage is inclined to the surface under investigation the hardness varies in opposite directions. In figure 285, for instance, the hardness is greater in the direction b to a than in the direction a to b. Curves of Hardness. A curve drawn on a plane surface in such a way as to express the differences in hardness exhibited by a crystal face in different directions is known as a curve of hardness. The relative hardness in different directions is represented by the relative lengths of straight lines passing through the center of the curve and terminating at both ends in its cir- cumference. Figures 286 and 287 are the curves of hardness for the cubic faces of crystals of halite (NaCl) and of fluorite (CaF 2 ), respect- ively. In the first figure the lines along the diagonals of the faces are longer than those parallel to their edges. This indicates that the hardness in the former direction is greater than that in the latter. The hardness along intermediate directions is inter- mediate in degree. In the case of fluorite the hardness is least in directions parallel to the diagonals of the faces and greatest in directions perpendicular to the cubic edges. The symmetry of curves of hardness always corresponds to the MECHANICAL PROPERTIES OF CRYSTALS 191 symmetry of the faces for which they are constructed. Con- sequently from the differences in hardness exhibited by sub- stances the symmetry of their crystals may be determined. FIG. 286. FIG. 287. Curves of hardness in cubic faces of halite (Fig. 286) and fluorite (Fig. 287.) Figure 288 shows the curve of hardness as determined by three observers on the basal plane of calcite, which is an hexagonal mineral crystallizing in rhombohedral hemihedral forms. Three planes of symmetry are perpendicular to the basal plane and FIG. 288. Curves of hardness on the basal plane of calcite as determined by three different investigators. Although varying in shape they all exhibit the same symmetry. (After Mutter}. these intersect one another at angles of 120. In the figure their positions are indicated by the lines drawn from the angles of the circumscribing triangle to the centers of the opposite sides. 192 PHYSICAL CRYSTALLOGRAPHY The three curves, though of different shapes, are all symmetrical with respect to these three lines. Had the mineral been holo- hedral the curves would have been symmetrical about six lines intersecting at 60. Density or Specific Gravity. The density or specific gravity of a body is the relation existing between its weight and the weight of an equal volume of water at a given temperature. In brief, the specific gravity of a substance is the weight of a cubic centimeter of its material in grams. Strictly considered, density is as much a chemical as a physical property of matter, for while its value depends partly upon the closeness with which a body's molecules are packed together, i.e., the number of molecules within a unit volume, it is affected also by the weights of these molecules and the weights of the molecules depend upon the number, the nature, and the arrangement of their component atoms. Since gravity acts upon a homogeneous mass of matter as a whole rather than upon its component parts independently, the result of its pull cannot be governed by the laws of symmetry. However, in the case of dimorphous substances (substances crystallizing in two different systems) the two forms will possess different specific weights. Calcite and aragoni.e, for instance, are both forms of CaCO 3 , but the specific gravity of calcite, which is hexagonal, is 2.714 and that of aragonite, which is orthorhombic, is 2 . 94. Methods of Determining Densities. Theoretically, the simplest method of determining the specific gravity of a solid body is (i) to weigh a small fragment suspended from one arm of a balance by a silken thread, (2) to immerse in water and weigh again, (3) to subtract the weight in water from the weight in air, and (4) to divide this remainder into the original weight in air. The reasons for these manipulations are known to all students familiar with the principles of elementary physics. If a fragment whose weight in air is 3.25 grams loses 1.25 grams when weighed in water, its density is 3.25-1-1.25 or 2.6. The Jolly Balance. The most convenient instrument for the rapid determination of approximate densities is the Jolly MECHANICAL PROPERTIES OF CRYSTALS 193 balance. This consists essentially of a spiral of wire fastened at the top to a movable arm (see Fig. 289). At the lower end it is provided with two little pans, one suspended beneath the other. The lower pan is kept always immersed to the same depth in water, while the other one hangs in the air. On the upright standard behind the spiral is a mirror on which is engraved or painted a scale of equal parts. The object whose density is to be determined is first weighed in the upper pan, then transferred to the lower pan and weighed again. The second weight subtracted from the first will give the weight of the water replaced by the fragment. The original weight divided by this yields the result desired. The method of using the Jolly balance is simple. Before each weighing the zero point of the instrument is fixed by glancing along the end of the wire at its reflection in the mirror. The division of the scale that lies exactly in the line of sight between the point on the spiral and its image in the mirror is the zero point for that determina- tion. A small fragment of the substance whose density is sought is then placed in the upper pan, care being taken to prevent the bottom of the pan from touching the water in which the lower pan is immersed usually the vessel containing the water must be lowered immediately upon placing the fragment in the upper pan. When the instrument comes to rest and the vessel holding the water is adjusted until the sur- face of the water is as high above the lower pan as it was originally, a second reading is made. The fragment is then transferred to the lower pan, and the level of the water is again adjusted. A third reading is now taken. If the first reading 13 FIG. 289. The jolly balance for approximate determinations of specific gravity. 194 PHYSICAL CRYSTALLOGRAPHY be subtracted from the second and third readings the results will give the relative weights of the fragment in air and in water. The difference between these two results divided into the first weight will yield the specific gravity sought. The Use of Heavy Solutions. Often it is more convenient to determine the density of small fragments of crystals by some other method than either one of those described above. This is always true in the case of very small fragments and of powders. For the determination of the specific gravity of these use is made of solutions of high specific gravity, which by the addition of water or of some other liquid may have their densities lowered. The powder or small fragment is thrown into the solution. If it floats, the density of the solid is less than that of the liquid. This is gradually diluted with the proper medium until the fragments remain suspended in the mixed fluids, neither rising nor sinking. When this occurs fragments and liquid are of the same specific gravity. The density of the liquid may then be determined by any one of the physical methods appropriate for this purpose. Naturally the use of heavy solutions is limited to those substances whose densities are less than their own. There are serveral solutions in common use for the determination of specific gravities. The one most frequently employed is a solution of mercuric and potassium iodides, known as the Thoulet solution. Its greatest density is 3.19. It will serve to determine the densities of substances lighter than 3.19, but, of course, cannot be used for heavier substances. Among the other heavy solutions often used for the same purpose may be mentioned the Klein solution (cadmium borotungstate in water) with a maximum density of 3.6, the Rphrbach solution (barium-mercury-iodide in water) with a density of 3.59, and the Retgers solution (thalium-mercuro- nitrate at 76) with a specific gravity of 5.3. FIG. 290. Sepa- rating funnel for use with solutions. MECHANICAL PROPERTIES OF CRYSTALS 195 Practical Uses of Specific Gravity Determinations. Fragments of substances that closely resemble each other in general appearance and in many of their physical and chemical proper- ties may often be easily distinguished from one another by their densities, hence the determination of the specific gravity of a substance under investigation will often lead to correct inferences concerning its nature and identity. For example, anorthite and albite are two feldspars that resemble each other very closely in appearance. The former, however, has a specific gravity of 2.76 and the latter of 2 . 63. Further, a mixture of substances in a finely divided form may be separated into its component parts by use of a heavy solution in a separating funnel, such as is represented in figure 290. The heaviest of the powders will fall and can be drawn off while the lighter ones remain floating on the solution. LIST OF SOME IMPORTANT MINERALS AND THEIR DENSITIES. Sulphur 2.05 Topaz 3- 56 Rock salt 2.10 Garnet 3.75 ** Gypsum 2 . 30 Celestite 3.97 Orthoclase 2 . 56 Rutile 4.25 Quartz 2 . 65 Magnetite 5 . 20 Calcite 2.72 Hematite 5 . 30 Muscovite 2.85 Cassiterite . 6 . 84 Biotite 3.01 Cinnabar 8 . oo METALS AND THEIR APPROXIMATE DENSITIES. Potassium 86 Iron 7 . 80 Magnesium ... i . 80 Copper 8 . 90 Aluminium 2.50 Silver 10.60 Zinc 7 . 10 Gold 19 . 30 CHAPTER XVIII. OPTICAL PROPERTIES OF CRYSTALS. Introduction. The light that falls upon a crystal is partly reflected from its surface, partly transmitted through its sub- stance, and partly absorbed within it. In either case the light suffers changes which affect its character, and the nature of these changes is determined by the material of the crystal and by its physical symmetry. Properties Dependent upon Reflection. Whenever a ray of light falls upon a surface a portion of it is reflected in accordance with the simple law : the angle of reflection is equal to the angle of incidence and both are in the same plane. Moreover, some of reflected light suffers a change which causes it to possess cer- tain properties not possessed by ordinary light. Because of the phenomena exhibited by this changed part of the reflected light, this part is described as being polarized; i.e., the vibrations that transmit it oscillate in a single plane parallel to the line of trans- mission. The properties depending upon the character of the reflected light are color and luster. Color. The color of a crystallized substance, like that of other substances, depends upon the character of the light reflected from its surface, which in turn depends upon the color of the light that falls upon it, upon the color and quantity of that which is absorbed, and upon the quality of the reflecting surface. When illuminated by light from different sources the same crystal often appears quite differently colored. The most remarkable case of this kind is the mineral alexandrite, or chryso- beryl (BeAl 2 O 4 ), the value of which as a gem depends upon the fact that it appears green by daylight and red by lamplight or gaslight. Indeed, the colors of substances vary so greatly in different lights that mineralogists have found it necessary to 196 OPTICAL PROPERTIES OF CRYSTALS 197 agree that the colors mentioned in their descriptions of minerals shall refer only to the colors in daylight. When wKite light falls upon any surface some portion of it is absorbed or transmitted, the remainder being reflected. If certain rays are absorbed or transmitted to a greater extent than certain others the reflected light will be of a different color from the incident light, for it will be made up of white light minus the rays that have been absorbed or transmitted. When white light falls upon a mass of cinnabar (HgS), for instance, certain of its rays are absorbed. Those that remain and are reflected consti- tute red light, and the mineral appears red. The color of opaque substances is much more characteristic than that of transparent or translucent ones. In the latter case the distinctive color (ideochromatic) is often obscured by im- purities or by some pigment (allochromatic) which is present in such small quantity that its nature in many instances cannot be determined. Hence, not much reliance can be placed in the colors of such substances as distinctive characteristics. For instance, many minerals that are colorless when pure are found in nature with a great variety of tints. Tourmaline is a case in point. While transparent colorless crystals are known, the predominant colors are black, brown, green, and red. Although opaque substances possess much more character- istic colors than do transparent or translucent ones, it frequently happens that their true colors are obscured by a surface tarnish. The true color is the color on a fresh fractured surface. Streak. Not only does the color of a substance vary with its nature, but it varies also with the character of its 'reflecting surfaces. More light is reflected from a smooth surface than from a rough one, and, conversely, more light is absorbed by a rough surface than by a smooth one. As a consequence, the light reflected from a smooth surface and that from a rough one may be made up of different components differently combined, and hence the colors of the two surfaces may appear different. A pyrite (FeS 2 ) crystal has a bright, brassy-yellow color, while a rough- surfaced fragment of the same mineral may appear bronzy- yellow or greenish. 198 PHYSICAL CRYSTALLOGRAPHY A still greater contrast in color is often noted when crystals of a substance are compared with their powders. This is especially true of opaque minerals and of translucent ones that are highly colored. While pyrite crystals are brassy-yellow, their powder is greenish-black. Hematite in crystals is often of a steel-black color. Its powder is blood-red or reddish-brown. The color -of the powder of a substance is more characteristic than that of its crystals, partly because in the powder the effect of variations in the quality of the reflecting surfaces is eliminated. The most convenient method of obtaining a small quantity of crystal powder is by drawing a fragment of the crystal across a rough porcelain surface. The mark left on the porcelain is known as the streak. In all descriptions of mineral species the color of the streak is mentioned. Transparent and translucent substances, whatever their color, usually possess a white streak. This is not so much due to differences in the absorbent power of large fragments and of their powder as it is to the dilution of the pigment that colors them. On the same principle a tumbler of water to which a little blue litmus solution is added has a bluish tint, while a single drop of the same water is practically colorless. Luster. The surfaces of crystals often present very decided peculiarities independently of color. Some surfaces reflect nearly all the light that falls upon them, others but a small quantity. The former glisten perceptibly, the latter are dull. Others reflect a portion of the light from their outer surfaces, and a portion from the surfaces of cracks, etc., within them. Some others scatter the light diffuse it. The result of these different phenomena is known as luster. The terms most frequently used in describing lusters are the following: metallic, vitreous, adamantine, resinous, pearly, and silky. The metallic luster is confined to opaque substances. It is the luster that is characteristic of the metals. The vitreous luster is that of glass; the adamantine, that of the diamond and of other very brilliant transparent substances; and the resinous, that of rosin. The pearly luster is. found only in those substances that have a very perfect cleavage or that are traversed by numerous OPTICAL PROPERTIES OF CRYSTALS 199 cracks. The play of colors that is its characteristic feature is caused by the interference of a portion of the light reflected from the sides of the cleavage or other cracks, a phenomenon analogous to that of Newton's rings. It is so called because it is the luster of satin and of silk. The Transmission of Light. That portion of incident light which is not reflected from the surface of a substance penetrates its mass, and either passes through it or is stopped by it. That portion which passes through is said to be transmitted. The por- tion that is stopped is said to be absorbed. Nearly all substances transmit some light, though often to such a slight degree that they are practically opaque. An opaque substance transmits no light, therefore it appears dark when viewed from the side opposite to that on which the light is incident. A transparent substance transmits nearly all the light that penetrates its mass; i. e., nearly all the light that is not reflected from its surface. When viewed from the side opposite to that on which light is falling, it may be white or it may possess a distinct tint. An object can be seen distinctly through a transparent substance. A translucent sub- stance transmits some light, but not enough to render distinctly visible an object viewed through it. A piece of iron is opaque, colorless glass is transparent, and porcelain translucent. Absorption of Light. The difference between opaque, transparent, and translucent substances is due to the difference in the quantity of light absorbed by them. The opaque bodies absorb all the light that is not reflected, the translucent ones absorb a considerable quantity of it, and the transparent bodies almost none. In many translucent and transparent bodies light of a certain color may be absorbed while that of a different color is transmitted. As a result of this phenomenon the substance may have a distinct color in transmitted light which is usually different from that produced by reflected light. Thin sheets of gold are yellow by reflected light and green by transmitted light. The color of the light transmitted may be determined by the nature of the sub- stance through which it passes or it may be determined by inclu- sions or by a pigment diffused through its mass. The former is 200 PHYSICAL CRYSTALLOGRAPHY an ideochromatic color, and the latter allochromatic. More- over, different portions of crystals may transmit light of different colors. This will happen when, during the crystal's growth, the solution from which it separated changed in composition from time to time, and consequently material of a slightly different character was successively added to the nucleus already in existence. The result is a distribution of color which is fre- quently zonal (see p. 150), as in quartz, tourmaline, augite or FIG. 291. Thin section of rock show- FIG. 292. Vertical section of ottrelile ing zonal structure in hornblende. in rock, showing regular distribution of Section cut parallel to the basal plane color. Magnified 50 diameters. (After of the hornblende. Magnified 50 Rosenbusch}. diameters. (After Rosenbusch). hornblende (Fig. 291), and sometimes disposed in accordance with the presence of certain crystal planes, as in ottrelile or augite (Fig. 292). Not only do different substances vary in their powers of absorp- tion, but the same substance when in crystals may possess this power in different degrees along different directions. Light transmitted in a certain direction through a crystal may thus lose some of its rays, while other rays may be lost when transmission occurs through some other direction. Thus a crystal may appear differently colored when viewed by transmitted light in different directions. Pleochroism. This is the general term applied to the prop- erty of exhibiting different colors in different directions. If a crystal possesses different colors when viewed in two different OPTICAL PROPERTIES OF CRYSTALS 2OI directions, it is said to be dichroic; if it exhibits three distinct colors when viewed in three different directions, it is trichroic. In the case of amorphous and isometric substances the same colored light is absorbed irrespective of the direction of the rays transmitted through them. These bodies thus exhibit no pleo- chroism. In crystals possessing a principal plane of symmetry the absorption in the direction of the c axis is often different from that which takes place in the direction of the lateral axes. Hence these crystals may be dichroic. A cube of dark tourmaline, an hexagonal mineral, when viewed through the direction of the c axis appears much darker than when viewed in the direction at right angles to this, and may possess different colors in the two directions. The absorption is greater for the ray transmitted parallel to c than for the rays transmitted parallel to the plane of the axes a. This fact is represented by the formula O > E. The symbol O refers to the ray transmitted in the direction of the c and E to the ray transmitted parallel to the principal plane of symmetry (see also page 205). Only crystals possessing no principal planes of symmetry may be trichroic, the absorption being different along three directions perpendicular to one another; consequently, these crystals may appear of three different tints when viewed along these three different directions, and at the same time they may possess different degrees of transparency in these same directions. The mineral cordierite, an orthorhombic mineral, is often strongly trichroic in light blue, dark blue and yellowish-white tints. Glaucophane, a monoclinic mineral, is blue, violet, and yellowish-green. Axinite, a triclinic mineral, is colorless, yellow, and violet. Nearly all transparent minerals are pleochroic in some degree, but frequently their differences in tint along different directions are so slight that they can be detected only by the aid of an instrument known as the dichroiscope, which consists simply of a small rhomb of iceland spar (calcite) mounted in a brass tube closed at both ends except for two small holes which serve as peep-holes, and through which the crystal is viewed. The little rhomb causes double refraction (see pp. 204-205) and thus separates the two rays. 202 PHYSICAL CRYSTALLOGRAPHY Relation Between Pleochroism and Crystal Symmetry. Only crystallized substances can exhibit pleochroism, and not all these exhibit it. Opaque crystals can, of course, not exhibit it. Moreover, crystals that are completely transparent in every direction cannot exhibit it. These are colorless in all directions because there is no absorption. Isometric crystals likewise absorb equally in all directions and consequently must show a single color by transmitted light. Pleochroism is therefore limited to anisotropic substances substances crystallizing in any system but the isometric because only in anisotropic sub- stances is light absorbed differently in different directions. Hexagonal and tetragonal crystals may show dichroism, but they cannot exhibit trichroism. Orthorhombic , monoclinic, and triclinic crystals may be trichroic* Fluorescence and Phosphorescence. Some substances such as fluorite (CaF 2 ) possess the property of transmitting light of a certain color while at the same time radiating light of some other color. This property is known as fluorescence. Fluorite may appear green by transmitted light, while at the same time it glows with a pale blue light. This property of fluorescing (from the name of the mineral best exhibiting it) is now known to be possessed by many crystals, such as those of the uranium minerals, and those containing fluorine and boron, and the manu- factured substances fuchsin (C 20 H 19 N 3 .HC1), fluorescein (C 20 H 12 - O 5 ), and magnesium cyanplatinite (MgPt(CN) 4 + 7 H 2 O). In the case of crystals of magnesium cyanplatinite and of some other substances it is noted that the fluorescence is different in color from different faces. In the cyanplatinite, which is tetragonal, the prismatic faces fluoresce green and the basal planes bluish-red. In general, similar planes fluoresce alike, dissimilar planes may fluoresce differently. Other substances, for example, phosphorus, possess the power of giving off light rays in the dark at comparatively low temper - * The mineral dealers furnish at low prices little plates of mica, beryl, cor- dierite, tourmaline, and other minerals so mounted in cork cells that they may readily be revolved about an axis. On holding the plates opposite a window and revolving them, their differences in color along different directions are conveniently studied. OPTICAL PROPERTIES OF CRYSTALS 203 atures. Many specimens of diamond (C), calcite (CaCO 3 ), etc., glow when removed from the presence of sunlight into a dark room. This property is known as phosphorescence. Other * minerals, such asfluorite (CaF 2 ) and apatite (Ca 5 P 3 O 12 Cl), glow brightly when heated to a temperature far below that of red heat. Others become luminous when their crystals are rubbed together, as, for instance, sugar and zinc-blende (ZnS), and others glow when subjected to the action of the cathode rays or of radium emana- tions. The difference in the be- havior of natural and imitation gems under the action of the cathode rays is sometimes made use of in detecting frauds. Refracted Light. If a ray of light traveling through air fall upon a transparent solid or liquid body obliquely, it suffers a change in direction in its passage through the body; i.e., it is refracted. Both the incident and the refracted rays are in the same plane, but the latter is bent toward the perpendicular to the surface of the body at the point of incidence of the former. For any given substance the amount of refraction or bending varies with the obliqueness of the incident ray, the refraction being greater as the angle of incidence becomes larger. If in figure 293 CB represent the path of a ray of light falling on the surface of a substance at B, its angle of incidence is C B A or i. This ray, upon entering the substance, suffers a change in direction, its new path being represented by B R. Its angle of refraction is r. The Index of Refraction. For every uncrystallized substance and every substance crystallizing in the isometric system there is always a definite relation existing between the angle of incidence and the angle of refraction. This relation is a constant one for light of a given color, provided the medium surrounding the body is the same in all cases. For example, the relation existing between the angle of incidence of a ray of light passing through air and 204 PHYSICAL CRYSTALLOGRAPHY striking a surface of glass obliquely and its angle of refraction in the glass is always the same no matter what may be the size of the angle of incidence. The ratio is expressed by the formula n = , in which n is the constant, i the angle of incidence, sin r and r the angle of refraction (see Fig. 293). The greater the amount of the bending, or refraction, of a ray, the smaller the angle r, hence the less the value sin r as compared with the value sin i, and the greater the constant n, or the greater the power of refraction of the body to which it refers. The ratio n, as has already been stated, is a constant for all rays of light of the same color entering a homogeneous, amor- phous, or isometric body from a given medium, as, for instance, air, irrespective of their obliquity. It differs, however, for different substances, and varies with the medium traversed by the rays before they enter the refracting substance, and also with the color of the light. It is because of the differences in amount of refraction suffered by rays of different colors that the spectrum is produced when white light is allowed to traverse a glass prism. When the constant n refers to the refraction of a ray of light of a definite color passing from a vacuum into a body, it is known as the index of refraction. It is characteristic for different sub- stances, and is often employed for identifying them. The indices of refraction for several substances are as follows: Water =1.33. Air =1.0003. Rock salt = i. 54. Diamond = 2. 4195. Crown Glass =1.53. Flint glass =1.61. Single and Double Refraction. In all amorphous trans- parent bodies and in all those substances that crystallize in the isometric system the index of refraction remains the same, what- ever the direction of the incident ray, provided the light is the same in color and the medium through which it passes is the same. Such bodies are known as isotropic or singly refracting. In them there is only one refracted ray. Light falling upon substances that crystallize in any other system than the isometric is usually split up by refraction into two rays OPTICAL PROPERTIES OF CRYSTALS 205 following different paths (Fig. 294). These bodies are known as aniso tropic or doubly refracting. Both of the refracted rays are polarized. An excellent example of a doubly refracting substance is a cleavage piece of calcite or iceland spar (CaCO 3 ), which on ac- count of its strong doubly refracting power has long been called " double-spar." If a plate of the clear mineral be placed over a pin-hole in a piece of cardboard and both be held up to the light, the pin-hole will appear double (Fig. 294). If the plate be now revolved parallel to the plane of the cardboard one of the pin- FiG. 294. Photograph of double image of pin-hole as seen through a cleavage rhombohedron of calcite. hole images will revolve about the other, which will remain fixed in position. The light ray that produces the fixed image may easily be shown to obey the law of simple refraction its index of refraction is a constant. This is called the ordinary ray. The ray that produces the other image obeys another law. It is called the extraordinary ray. Its index of refraction varies with the direc- tion pursued by the transmitted ray. In calcite the index of refraction of the ordinary ray (written o>) is i . 6585 for yellow light, while that of the extraordinary ray varies between i . 6585 and 1.4863. The value that differs most from to is taken as the 206 PHYSICAL CRYSTALLOGRAPHY index of refraction for the extraordinary ray (written e). In this instance w > e. The mineral is said to be optically negative. In quartz, on the other hand, C( >C< )Ca, \ o / \ o / \ o / \ o / for aragonite. Polymorphous modifications of a substance are the result of variations in the conditions under which its different forms are produced. These differ in purely physical properties as widely as they do in geometrical properties. Orthorhombic sulphur crystallizes from solution. Monoclinic sulphur is obtained by cooling a fused mass. The one is transformed into the other at about 96. Orthorhombic sulphur melts at 113.5, and monoclinic sulphur at 119.5. Tne density of the former is 2 . 05 and of the latter i . 96. Mercuric iodide is another substance possessing well-known modifications. From solution it crystallizes as red tetragonal crystals; but from a fused mass it separates as yellow orthorhom- bic crystals. The red variety passes into the yellow variety when it is heated to 126. PARTIAL LIST OF POLYMORPHOUS BODIES. Dimorphs. Pyrite (regular) FeS 2 (orthorhombic) Marcasite Arsenolite (regular) As 2 O 3 (monoclinic) Claudetite Calcite (hexagonal) CaCO 3 (orthorhombic) Aragonite Yellow (orthorhombic) HgI 2 (tetragonal) red ISOMORPHISM AND POLYMORPHISM 229 Trimorphs Ti0 2 Anatase (tetragonal, a : c=i : .71) Rutile (tetragonal, a : c=i : .64) Brookite (orthorhombic) Sp.Gr. = 3. 84. Double Re- fraction . Sp.Gr. = 4.24. Double Re- fraction + . Sp.Gr. =4.06. Double Re- fraction + . Al 2 SiO 5 Cyanite (triclinic) Sp.Gr. = 3.60. Hardness 5-7. Sillimanite (orthorhombic) Sp.Gr. = 3.24. Hardness 6-7. Andalusite (orthorhombic) Sp.Gr. = 3.18. Hardness 7. Quartz (hexag.) Tridymite (orthorh.) Asmanite (tetrag.) Tetramorph. Sulphur may be orthorhombic, monoclinic with a : b : c = .996 : i : .999, /3=9546 / , monoclinic with a :b :c=i .06 : i : .71, ,9= 91 47', and hexagonal. Isomorphous Compounds. A comparative study of crystal- lized bodies has shown that those possessing analogous composi- tions often possess also the same general crystalline form. Crystals are of the same general crystalline form when they possess the same grade of symmetry and the same habit and have their corresponding interfacial angles of nearly the same value. Compounds possess analogous compositions when their for- mulas are of the same type, as Na 2 CO 3 and K 2 CO 3 or Na 3 PO 4 , Na 3 AsO 4 and K 3 AsO 4 . Analogous compounds may be regarded as derived from one another or from some common source by the replacement of single elements or groups of elements by certain other nearly allied elements or groups of elements. For instance, by replace- ment of the hydrogen in H 2 CO 3 the following series of compounds may be derived: MgCO 3 , CaCO 3 , FeCO 3 , MnCO 3 , etc. These possess analogous compositions. 230 CHEMICAL CRYSTALLOGRAPHY Isomorphous bodies, in brief, are those possessing similar crys- tal forms. Isomorphism, or the property of being isomorphous, is limited to bodies of analogous compositions. The following short list of isomorphous minerals will illus- trate the meaning of the term: PARTIAL LIST OF ISOMORPHOUS GROUPS. Orthorhombic Hexagonal Orthorhombic Forsterite Mg 2 SiO 4 Calcite CaCO 3 Diaspore AIO(OH) Tephroite Mn 2 SiO 4 Magnesite MgCO 3 Goethite FeO(OH) Fayalite Fe 2 SiO 4 Spherocobaltite CoCO 3 Manganite MnO(OH) Siderite FeCO 3 Rhodochrosite MnCO 3 Smithsonite ZnCO 3 Morphotropism. Although isomorphous compounds possess the same general crystallographic habit, there are no two of them exactly alike. With a change in the chemical composition of any substance there is a corresponding, though sometimes but slight, change in its morphological and physical properties. That partial change which is effected in the crystallization of a substance by the replacement of one of its constituents by some other element or group of elements is known as morphotropism. This usually consists in a slight change in the axial ratio of its crystals and a correspond- ing change in the values of their interfacial angles. The change in morphological properties is, of course, attended with changes in physical properties. The character and amount of morphotropic change produced by the introduction of an element, or group of elements, into any compound is known as its morphotropic action. The variation in axial ratio shown by members of two iso- morphous groups is illustrated below: Hexa gonal-He mihedral Orthorhombic Calcite, CaCO 3 , a:c= 10.8543 Mg 2 SiO 4 , a : b : c= .4666 : i : . 5868 Rhodochrosite, MnCO 3 , a :c = 10.8259 Mn 2 SiO 4 , a : b : c= .4621 : i : .5914 Siderite, FeCO 3 , a:c= : 0.8191 Fe 2 SiO 4 , o : 6 :c= .4584 : i : .5791 Magnesite, MgCO 3 , a : c = : 0.8095 Smithsonite, ZnCO 3 , a:c= : 0.8062 Isomorphous Mixtures. Since isomorphous compounds have the same crystallographic form, they naturally tend to crystal- ISOMORPHISM AND POLYMORPHISM 231 lize together when present in the same crystallizing solution. The result of this crystallization is a mixture of the different compounds constituting individual crystals, which to all tests appear homo- geneous throughout their entire masses when the volume of the crystallized substance that separates is small as compared with the volume of the solution. Mixtures of this kind are called iso- morphous mixtures. They are very common among minerals, perhaps more common among the silicates than are simple compounds. Isomorphous compounds may be defined from this point of view as those which may crystallize together or which may unite in various proportions to produce homogeneous crystals. The proportions of the substances in the mixed crystals bears no fixed relation to their molecular weights, as is the case in molecular compounds. On the other hand, they may occur in practically any proportion, depending upon the composition of the solution from which the crystals separate. In some cases there is manifested a tendency for the substances to crystallize together in certain proportions rather than in others, but in many other cases they may crystallize in all proportions possible. The property of forming mixed crystals is regarded as so characteristic a feature of isomorphous substances that substances are not generally regarded as isomorphous until they have been made to crystallize together or have been found in crystals in various proportions which are not in the ratio of their molecular weights. Illustration of Properties of Isomorphous Mixtures. An instructive illustration of the dependence of the physical and mor- phological properties of substances upon their chemical composi- tion is afforded by the crystals composed of mixtures of MgSO 4 + 7H 2 O and Zn SO 4 +7H 2 O. These compounds are known as epsomite and goslarite when found in nature. They crystallize in the sphenoidal division of the orthorhombic system, and they form mixed crystals containing all proportions of the two molecules. The properties of the crystals are determined by the percentage of the magnesium (or the zinc) salt present in them. The following table exhibits these relations for a few of the mixtures investigated : TOO i . 6760 8 9 25' 74-44 1.7472 89 15' 57-59 1.7977 89 8' 35-64 i . 8604 89 i' i8.ii i . 9094 ooo ^ ./ as 54 i . 9600 88 48' 232 CHEMICAL CRYSTALLOGRAPHY PROPERTIES OF ISOMORPHOUS MIXTURES OF MgSO 4 .7H 2 O and ZnSO 4 .7H 2 O. Per cent, of Mg Molecule Sp. Gr. Angle ooP/\ ooP Optical Axial Angle 78 18' o" 76 42' 47"* 74 3' 45"* 70 53' In general the physical properties of crystals composed of mixtures of isomorphous compounds are functions of the com- pounds crystallizing together, being intermediate between the two pure compounds. This relationship between the physical properties and the chemical composition of mixed crystals is so close that Retgers declares that " Two substances are truly isomor- phous only when the physical properties of their mixed crystals are continuous functions of their chemical composition" Formulas of Isomorphous Mixtures. Although isomor- phous mixtures are not definite combinations of elements in the same sense as are simple compounds, nevertheless, they exist so frequently that some means must be decided upon for the representation of their composition. The logical method of writing their formulas is to designate the number of molecules of each of the substances entering into the mixture. This method, however, would require a different formula for every different mixture possible. Since the number of these is often large, a great number of formulas would be demanded, and some of them would be very complicated. In practice it is deemed sufficient to indicate by the formula the nature of the molecules in the mixture without specifying exactly their proportions. This is done by enclosing the symbols of the mutually replaceable elements or groups of elements in parentheses with a comma between them, and writing the symbols of the remaining elements in the usual manner. Such a formula, as, for instance, (Fe,Mg) 2 SiO 4 , indicates that iron and magnesium *Approximate. ISOMORPHISM AND POLYMORPHISM 233 silicates are present in different proportions in a series of com- pounds possessing the same general crystallographic and physical features. ILLUSTRATIONS OF FORMULAS OF ISOMORPHOUS MIXTURES. Spinel is MgAl 2 O 4 . Iron spinel is MgFe 2 O 4 . An isomor- phous mixture of these is represented by Mg(Fe,Al) 2 O 4 . Barite=BaSO 4 . Celestite = SrSO 4 . Barito-celestite = (Ba,Sr) SO 4 . Tetrahedrite=(Ag 2 ,Cu 2 ,Fe,Zn,Hg) 4 )(Sb,As) 2 S 7 . Isodimorphous Groups. A series of isomorphous poly- morphs is an isopolymorphous group. Series of isomorphous compounds each one of which is a dimorph are very common. Such groups are known as isodimorphous groups. A simple example is the following: Regular. Orthorhombic. Arsenolite, As 2 O 3 Claudetite Senarmonite, Sb 2 O 3 Valentinite Double Salts. Many crystallized substances that are not isomorphous mixtures appear to consist of a combination of mole- cules which, however, unite in definite proportions, and not in many different proportions like the molecules in isomorphous mixtures. They are known as double salts because they appear to be made up of portions that may exist independently as simple compounds. An illustration in case is sodium silver chloride, whose formula is NaCl +AgCl, or cryolite, a monoclinic mineral with the composition Na 3 AlF 6 or 3NaF.AlF 3 . Substances con- taining water of crystallization also belong to this class of mo- lecular compounds. Gypsum, CaSO 4 +2H 2 O, is an example. It is not at all certain that the double salts differ in any essen- tial respect from ordinary atomic molecules. They are referred to here only for the purpose of emphasizing the fact that isomor- phous compounds form mixed crystals in which the proportions of the components present may vary with varying conditions dur- ing growth, whereas the crystals of a double salt the components of which are not members of an isomorphous series have a defi- nite composition which is invariable. INDEX. Absorption of light, 199-202 Acicular, 134 Aggregates, 143^147 botryoidal, 146-147 cryptocrystalline, 144 crystalline, 143, 144-147 fibrous, 146 globular, 145 granular, 145 lamellar, 145 radial, 145 sheaf-like, 147 Allochromatic, 197, 200 Amorphous substances, 6, 160, 177, 180, 214 Analogue pole, 216 Antilogue pole, 216 Axes, crystallographic, 17 of hexagonal system, 55-56 of isometric system, 32-33 of monoclinic system, 116 of orthorhombic, 102-103 of regular system, 32-33 of tetragonal system, 88 of triclinic system, 125 projection of, 169-173 Axial angle, 207-208 apparent, 208 true, 207 ratio, determination of, in hex- agonal system, 57-60 determination of, in tetragonal system, 89-90 of hexagonal system, 56-57 of monoclinic system, 116-117 of orthorhombic system, 103104 of tetragonal system, 89-90 of triclinic system, 1 26 Axis, twinning, 152 B Basal pinacoid, 64, 119 Biaxial crystals, 206 Boiling point, 211 Botryoidal, 146, 147 Brachyaxis, 103, 126 domes, 108, 128 hemi domes, 128 hemiprisms, 127-128 pinacoid, 109, 128-129 prisms, 107-108, 127-128 pyramid, 107-108, 126-127 series, 107-108 tetrapyramids, 127 Chemical compounds, 227 Classes of symmetry: dihexagonal bipyramidal, 60-69 ditetragonal bipyramidal, 87-95 ditrigonal bipyramidal, 78-80 ditrigonal pyramidal, 80-81 domatic, 121-122 dyakisdodecahedral, 46-48 hexagonal bipyramidal, 75-76 hexagonal trapezohedral, 76-77 hexoctahedral, 31-38 hextetrahedral, 48-51 orthorhombic bipyramidal, 105110 orthorhombic bisphenoidal, 112-114 orthorhombic pyramidal, 110-112 pentagonal icositetrahedral, 46 pinacoidal, 125-131 prismatic, 117-121 sphenoidal, 122-124 tetragonal bipyramidal, 98-99 tetragonal scalenohedral, 95-98 tetragonal trapezohedral, 100 trigonal scalenohedral, 70-74 trigonal trapezohedral, 82-86 235 2 3 6 INDEX. Cleavage, 181-183 planes of, 181-182 symmetry of, 182-183 Clinoaxis, 116 domes, 119 hemipyramids, 118-119 pinacoids, 119-120 prisms, 119 Closed forms, 68, 109 Coefficient of elasticity, 180 of expansion, 210-211 Cohesion, 181 Colloids, 6, 8, 1 60 Color, 196-197 allochromatic, 197,200 ideochromatic, 197,200 Columnar, 134 Combinations, 43-44 hexagonal, 68-69, 73-74, 80-8 1, 85-86 isometric, 38-39, 51-53 monoclinic, 120-121 orthorhrombic, 108, 109-110 tetragonal, 94-95, 99-100 triclinic, 129-130 Composition face, 152 Conchoidal fracture, 187-188 Conduction of electricity, 215 of heat, 209-210 Congruent forms, 47 Constancy of interfacial angles, 14-16 Contact twins, 152-153 Contraction in crystals, 210-211 Corrosion of planes, 136 Cryptocrystalline, 144 Crystal angle, 10 axes, 17 biaxial, 206 definition of, 8 drawing, 168-174 edge, 10 form, 21, 27 group, 143 inclusions, 141 individual, 8 liquid, 8 particles, 4-5 projection, 164-174 Crystal, structure of, 5-7 uniaxial, 206 Crystalline bodies, 6-9, 177 Crystallites, 141 Crystallization, 8-9 Crystallographic axes, 17 constants of monoclinic system, 116-117 of triclinic system, 1 26 Crystallography, 9 definition of, xi laws of, 9-10 systems of, 28, 29-30 comparison of, 130-131 Cube, 37 Curvature of planes, 136 Cyclic twins, 155 Dana's notation, 22 Deformed crystals, 134-135 Dendritic growth, 149 Density. See specific gravity. Determination of axial ratio in tet- ragonal system, 89-90 in hexagonal system, 57-60 Diamagnetic, 214 Diathermous, 209 Dichroiscope, 201 Dichroism, 201 Dihexagonal bipyramid, 61-62 prism, 64 series, 62-64 Dimorph, 162 Dimorphous substances, 227-229 Diploid, 47 Distorted crystals, 132-133 Ditetragonal bipyramid, 90-91 series, 91-92 Ditrigonal bipyramids, 79-80 prisms, 79-80 of second order, 84 Dodecahedron, 36 Double refraction, 204-206 explanation of, 205-206 Double salts, 233 Drawing of crystals, 168-174 INDEX. 2 37 Druse, 149-150 Dyakisdodecahedron, 47 Elasticity, 180 Electrical properties, 214-218 Electricity, conduction of, 215 Enantiomorphous forms, 83 Etched figures, 136, 220-223 symmetry of, 221-223 Expansion of crystals, 210-211 Extraordinary ray, 205 Fibrous aggregates, 146 Fluid inclusions, 139, 142 Fluorescence, 202 Fossilization, 163 Fourlings, 154, 157 Fracture, 186-187 Fusibility, scale of, 212-213 Fusing points, 211 table of, 212 Fusion, 2ii 212 Gas inclusions, 139-142 Glass inclusions, 141 Gliding planes, 183-184 Globular aggregates, 145 Goniometer, 11-13 Granular aggregates, 145 Groundform, 33 Gyroidal hemihedrism, 46 H Habit of crystals, 14, 133-134 acicular, 134 columnar, 134 prismatic, 134 tabular, 134 Hardness, 187-192 curves of, 190-191 determination of, 189 differences in, 189, 190 list of relative, 188 scale of, 187 Hauy, x Heat, conduction, 209-210 transmission, 209 Heavy solutions, 194 Klein, 194 Retgers, 194 Rohrbach, 194 Thoulet, 194 Hemibrachydomes, 128 Hemiclinodomes, 122 Hemihedrism, 31, 342 in hexagonal system, 69-82 in isometric system, 41-53 in monoclinic system, 123124 in orthorhombic system, 112-114 in tetragonal system, 95-101 law of, 42-43 Hemimacrodomes, 128 Hemimorphism, 31, 41-42 in hexagonal system, 80-8 1 in monoclinic system, 121-123 in orthorhombic system, 110-112 in tetragonal system, 101 Hemiprisms, 122, 127-128 Hemipyramids, 118 Hexagonal prisms of first order, 65-66 of second order, 66-67 pyramids of first order, 65-66 of second order, 66-67 rhombohedrons, 71-73 scalenohedrons, 71 Hexagonal system, 5486 axes of, 55-56 axial ratio in, 56-57 determination of, 57-60 combinations in, 68-69, 73-74 comparison with tetragonal system, 54-55, 87 general form in, 61 groundform in, 56 hemihedral division, 69-82 hemimorphism in, 80 8 1 holohedral division, 60-69 symmetry of, 54-55 tetartohedrism in, 82-86 three series of holohedrons in, 64-65 trapezohedrons, 77 Hexoctahedron, 34 Hextetrahedron, 49 2 3 8 INDEX. Holohedron, 31 Hopper- shaped crystals, 137 K Klein solution. 194 Icositetrahedron, 35 Ideal forms, 132 Ideochromatic, 197, 200 Idiomorphic form, 3, 4 Imperfections in crystals, 132-142 Inclusions, 138-142 crystal, 141 fluid, 139, 142 gas, 139, 141, 142 glass, 140 Index of refraction, 203-204 Indices, 23 Intercept on lateral axes, hexagonal system, 58-60 Interfacial angles, 10, n constancy of, 14-16 measurement of, 11-13 edge, 10 Interpenetration twins, 153, 154, 156, 157, 158 Irregularities in crystals, 132 Isodimorphous group, 233 Isometric system, 31-53 axes of, 32-33 combinations in, 38-39, 43-44 derivation of forms in, 34 determination of forms in, 37-38 general form in, 33-34 groundform in, 33 hemihedral division, 41-53 holohedral division, 31-40 symmetry of 31-32 tetartohedrism in, 53 Isomorphism, 227, 229-230 Isomorphous compounds, 230231 formulas of, 232-233 properties of, 231-232 groups, 230 mixtures, 230-231 Isotherms, 210 J Jolly balance, 192, 193 Lamellar aggregates, 145 Law of constancy of interfacial angles, 14-16 of hemihedrism, 4243 of rationality of the indices, 17 of simple mathematical ratios, 17-20 application of, 22 of symmetry, 28 of tetartohedrism, 42-43 Light, absorption of, 199 reflection of, 196 refraction of, 203-208 transmission of, 199 Liquid crystals, 8 List of fusing points, 212 of hexagonal hemihedrons, 81 of isometric hemihedrons, 51 of isodimorphous groups, 230 of isomorphous groups, 230 of polymorphous bodies, 228-229 of specific gravities, 195 of tetragonal hemihedrons, 101 Luster, 198, 199 M Macroaxis, 103, 126 domes, 108109, 128 pinacoid, 109, 128 129 prism, 107, 127-128 pyramids, 107, 126-127 series, 106 tetrapyramids, 127 hemiprisms, 127-128 hemidomes, 1 28 Magnetic properties of crystals, 214 Magnetism, 214 Mechanical properties of crystals, 180- *95 Merohedrism, 41 Microlites, 141 Miller's system of notation, 2324 Mimicry, 158 Mineral, definition of, ix INDEX. 239 Mineralogy, history of, x purpose of, ix subdivisions of, xi-xii Mohs's scale of hardness, 187 Molecules, 4-6, 7-8 Monoclinic domes, 19 hemiclinodomes, 122 hemipyramids, 118-119 orthodomes, 119 pinacoids, 119- 120 prisms, 119 system, 115-124 axes of, 116 combinations in, 120-121 crystallographic constants in, 116-117 groundforms in, 116117 hemihedral division, 123-124 hemimorphism in, 121-123 holohedral division, 117-121 symmetry of, 115-116 tetraorthodomes, 122 tetrapyramids, 122 Morphological mineralogy, 9 Morphotropism, 16, 230 N Naumann's system of notation, 21-22, 24 Negative crystals, 139 Notation, crystallographic, 20 Miller's system of, 23-24 Naumann's system of, 21-22, 24 Weir's system of, 20-21 Octahedron, 33, 37 Open forms, 68 -Optical axes, 207 properties of crystals, 196-208 Ordinary ray, 205 Orthorhombic brachydomes, 108-109 pinacoids, 108-109 prisms, 107-108 pyramids, 107-108 macrodomes, 108-109 pinacoids, 108-109 prisms, 106-107 Orthorhombic pyramids, 106-107 system, 102-114 axes of, 102-104 closed forms in, 109 combinations in, 108, 109-110 general form in, 105 groundform in, 103 hemihedral division, 112-114 hemimorphism in, iio-ni holohedral division, 105-112 symmetry of, 102 three series of forms in, 105 unit prisms, 105-106 prisms, 105-106 Parallel growths, 143, 147-151 Paramagnetic substance, 214 Parameters, 18-19 Paramo rphs, 162 Partial forms, 41 Parting, 186 Penetration twins, 153, 154 Pentagonal dodecahedrons, 47-48 hemihedrons, 46-48 icositetrahedron, 46 Percussion figures, 185, 186 Phanerocrystalline, 144 Phosphorescence, 202-203 Physical agencies, 178-179 Physical symmetry, 177, 178 Plane of symmetry, 28 Pleochroism, 200-202 Polymorphism, 227-229 Polymorphous bodies, 227, 229 Polysynthetic twins, 155, 156 Pressure figures, 186 Projection, crystal, 164-174 linear, 164-167, 168 of axes, 169-173 isometric, 170 tetragonal, 170 Orthorhombic, 170-171 monoclinic, 171 triclinic, 171-172 hexagonal, 172-173 spherical, 168, 167 240 INDEX. Pseudomorphism, explanation of, 161 Pseudomorphs, 160-163 chemical, 162 mechanical, 162-163 Pyramidal hemihedrism, 70, 75-76, 95, 98-99 Pyroelectrical properties of crystals, 216- 218 Pyroelectricity, 216 Pyritohedron, 47-48 R Radial aggregates, 145 Re-entrant angle, 152 Reflection of light, 196 Refraction, double, 204-206 single, 204 of light, 203-208 index of, 203-204 Relation between hexagonal pyramids and prisms of various orders, 67-68, 76 tetragonal pyramids and prisms of various orders, 94 Repeated twins, 154-158 Retgers solution, 194 Rhombohedral hemihedrism, 70-74 Rhombohedrons, 71-72 short symbols of, 73 Rock, definition of, ix Rohrbach solution, 194 Scale of fusibility, 212-213 hardness, 187 Scalenohedrons, 71, 96-97 Secondary twinning, 184 Sheaf-like aggregates, 147 Short symbols of rhombohedra and scalenohedra, 73 Simple mathematical ratios, 17 Simple refraction, 204 Sklerometer, 189 Solution, 219-223 planes of, 219-220 rates of, 219 symmetry of, 220 Specific gravity, 192-195 determination of, 192-195 list of, 195 use of, 195 Sphenoidal hemihedrism, 95-98, 112-114 Sphenoids, 97-98, 113-114 Stalactites, 3-4, 147 Streak, 197 Striations, 135 -136, 137-138 Structure of crystals, 5-7 Sub-individuals, 149 Supplementary twins, 153-154 Symbols of planes, 18-24 Symmetry, 25-30 axes of, 25 classes of, 29 grades of, 28 in crystallography, 26-28 law of, 28 of crystal faces, 178 of etched figures, 221-223 of pyroelectrical properties, 216-218 of solution, 220 physical, 177-178 planes of, 28 with respect to lines, 25-26 to planes, 25 to points, 26 Systems, crystallographic, 28, 29-30 Table of relative hardness, 188 of fusing points, 212 of hexagonal hemihedrons, 81 of isometric hemihedrons, 51 of tetragonal hemihedrons, 101 Tenacity, 180-181 Tetartohedrism, 31, 42 in hexagonal system, 82-86 in isometric system, 42, 53 in tetragonal system, 101 law of, 42 -43 Tetrabrachypyramids, 126-127 Tetragonal bipyramids of first order, 9 2 -93 of second order, 93-94 of third order, 98-99 INDEX. 241 Tetragonal prisms of first order, 92-93 of second order, 93-94 of third order, 98-99 scalenohedrous, 96-97 series of various orders, 92 sphenoids, 97-98 system, 87-101 axes of, 102-104 axial ratio in, 89-90 determination of, 89-90 comparison with hexagonal sys- tem, 54-55. 8 7 groundform in, 88-90 hemihedral division, 95-101 hemimorphism in, 101 holohedral division, 87-95 relations between forms of differ- ent orders in, 94 tetartohedrism of, 101 symmetry of, 87-88 Tetragonal trapezohedrons, 100 tristetrahedrons, 50 Tetrahedral hemihedrism, 48-51 Tetrahedrons, 50-51 Tetrahexahedrons, 36 Tetramacropyramids, 126-127 Tetramorph, 229 Tetra unit pyramids, 126-127 Thermal properties of crystals, 209-213 Thermoelectrical properties of crystals 215-216 Thoulet solution, 194 Transfer of Miller and Maumann symbols, 23-24 Trapezohedral hemihedrism, 70, 76-77, 95> I0 tetartohedrism, 80-81 Trapezohedrons, 83, 100 Trichroism, 201 Triclinic hemidomes, 127-128 hemiprisms, 127-128 pinacoids, 128-129 system, 125-130 axes of, 125 combinations in, 129-130 Triolinic system, crystallographic con- stants in, 1 26 groundform in, 126 symmetry of, 102 tetrapy ramids, 1 26 1 27 Trigonal bipy ramids of first order, 78-79 of second order, 84-85 hemihedrism, 78-80 prisms of first order, 78-89 of second order, 84-85 trapezohedrons, 83 tristetrahedrons, 49, 50 Trillings, 154, 156, i57> J 58 Trimorphs, 227, 229 Trisoctahedron, 36 Twinned crystals, 143 lamellae, 155 Twinning, artificial, 184 axis, 152 plane, 152 secondary, 184 Twins, contact, 152-153 cyclic, 155 interpenetration, 153, 154, 156 penetration, 153, 154 polysynthetic, 155, 156 repeated, 154-158 secondary, 184 supplementary, 153^154 U Uniaxial crystals, 206 V Vaporization, 211-212 Von Kobell, 212-213 W Weiss, x, 21-22 Weiss' system of notation, 21 Werner, x Zonal axis, 10 growths, 150, 200 Zone, 10 .10 14 DAY USE RETURN TO DESK FROM WHICH BORROWED EARTH SCIENCES LIBRARV This book is due on the last date stamped below, or on the date to which renewed. 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