UC-NRLF LECTURE NOTES ON CRYSTALLOGRAPHY BY HORACE BUSHNELL PATTON, Ph. D. $ Professor of Geology and Mineralogy at the State School of Mines, Golden, Colorado. 7EB1 v- PUBLISHED BY THE AUTHOR, GOLDEN, COLORADO. COPYRIGHT, 1896, BY HORACE B. PATTON. NEWS PRINTING COMPANY, 1896. PREFACE. The difficulty of presenting the subject of crystallogra- phy introductory to a course in mineralogy, with only a limited time at one's disposal, has been realized by all instructors in this branch. Even the best text books fail to present the subject in so clear a light as to do away with the necessity of lecturing. The main difficulty with the best treatises on this subject, when put in the hands of the student, is their length .and multiplicity of details. These lecture notes are not intended to form a treatise on crystallography. They were originally prepared, not for publication, but for use in the class room, supplemented by lectures, crystal models and natural crystals. In the very limited time allotted to the study of mineralogy it was found ERRATA. PAGE 8. Reverse the two signs >. < . in the table of forms. PAGE 11. Between the second and third lines of italics at the top insert all the faces that lie wholly within. PAGE 11. Center of page. Should read Trisoctahedron gives Tetra- gonal tristetrahedron. alogy. On account of the brevity of these notes the interleaved blank pages have been found very useful, and they will undoubtedly prove serviceable in case any one else should wish to use these notes as a basis for instruction. STATE SCHOOL OF MINES, GOLDEN, COLO., April, 1896. NEWS PRINTING COMPANY, PREFACE. The difficulty of presenting the subject of crystallogra- phy introductory to a course in mineralogy, with only a limited time at one's disposal, has been realized by all instructors in this branch. Even the best text books fail to present the subject in so clear a light as to do away with the necessity of lecturing. The main difficulty with the best treatises on this subject, when put in the hands of the student, is their length .and multiplicity of details. These lecture notes are not intended to form a treatise on crystallography. They were originally prepared, not for publication, but for use in the class room, supplemented by lectures, crystal models and natural crystals. In the very limited time allotted to the study of mineralogy it was found necessary to omit everything not absolutely essential to a clear and logical presentation of the subject of crystallogra- phy. Requests for copies of these lecture notes by other parties has led the author to hope that they may prove of service to some outside his own class room. No attempt is here made to enter upon many features of crystals, such as physical and optical properties, crystal aggregates, twinning, etc. These omissions are made for the reason that such features are usually clearly presented in almost any miner- alogy. On account of the brevity of these notes the interleaved blank pages have been found very useful, and they will undoubtedly prove serviceable in case any one else should wish to use these notes as a basis for instruction. STATE SCHOOL OF MINES, GOLDEN, COLO., April, 1896. CRYSTALLOGRAPHY. In the inorganic world bodies are met with that are homogeneous throughout; others, again, that consist of two or more kinds of homogeneous substances ; and still others that are composed of many distinct grains or particles of one kind of substance. The first are termed Minerals, the last two Rocks. A Mineral, therefore, may be defined as an inorganic, natural, homogeneous body. A mineral represents a definite chemical compound, and its formation is controlled by the same laws that con- trol the formation of all chemical compounds. These natural laws of growth express themselves in many ways, e. g., through outward form and through vari- ous physical and optical properties of the mineral. These different manifestations, however, vary with and thus char- acterize the chemical combination. As an illustration of these properties we may take what is called cleavage. Just as a block of wood cleaves in cer- tain directions, dependent on the grain of the wood, i. e., on the arrangement of the wood cells, so, many minerals are said to cleave in certain definite directions. And this min- eral cleavage is likewise due to a sort of grain in the min- eral produced by the arrangement of the molecules. To illustrate, rock-salt cleaves always in three directions at right angles to each other, while crystallized carbonate of lime cleaves in three directions that make an angle of about 105 to each other. Another, and more important, outward manifestation of the laws of mineral growth is seen in the natural external form. There are three ways in which minerals are usually formed, namely, through solution, through fusion and through sub- limation. But in any case, unless interfered with by some external agency, the mineral particles in forming are usually found to be bounded on all sides by plane surfaces and to 4 assume definite polygonal forms, which forms vary with the chemical composition of -the mineral, but are constant for any one definite mineral species. A mineral thus bounded entirely or partly by natural plane surfaces is termed a CRYSTAL. Crystallography, as far as these lecture notes are con- cerned, may be defined as a study of the geometrical rela- tionships of these natural crystal faces. Before taking up the study of crystal forms it is neces- sary to have clearly in mind a few fundamental DEFINITIONS. A COMMON SYMMETRY PLANE is any plane that divides a crystal body into two symmetrical halves; and a body may be said to be symmetrically divided when the following test holds true. Every perpendicular to the dividing plane, if extended in both directions to the surface of the crystal, must come out in points equally distant from the plane and simi- larly located on the crystal. If the symmetry plane were conceived to be replaced by a mirror, the image reflected by the half-crystal in front of the mirror would appear to coincide in position with the half-crystal behind the mirror. A SYMMETRY Axis is. any line (or the direction} perpen- dicular to a symmetry plane. A PRINCIPAL SYMMETRY PLANE is a plane in whicJi lie two, sometimes three, interchangeable symmetry axes. (These interchangeable symmetry axes are not the axes of this plane, but of other symmetry planes at right angles to this plane). Two axes are said to be interchangeable when, by turn- ing the crystal, you can bring one axis into the position oc- cupied by the other without thereby producing any appar- ent change in the position, form or direction of the crystal. A PRINCIPAL SYMMETRY Axis is any line (or the direc- tion) perpendicular to a principal symmetry plane. FACES. A crystal is bounded by plane surfaces called faces. EDGES. The line of intersection between two faces is called an edge. ANGLES. These are of two sorts. An inter facial angle is the plane angle formed by the intersection of two faces. A crystal angle is the solid angle formed by the intersection of three or more faces. SIMILAR EDGES AND ANGLES are those formed by the intersection of the same number of planes, similarly placed. Upon bringing together all known crystal forms it is seen that they may all be grouped into six crystal systems that differ from each other by the number and kind of the symmetry planes present. Below will be found grouped in accordance with the presence of principal and common symmetry planes the SIX CRYSTAL SYSTEMS. I. WITH 3 PRINCIPAL SYMMETRY PLANES. (1) with six common symmetry planes. II. WITH 1 PRINCIPAL SYMMETRY PLANE. (2) a, with four common symmetry planes arranged in two pairs of perpendicular planes. (3) b, with six common symmetry planes arranged in two groups of three each, making angles of 60 to each other. III. WITH NO PRINCIPAL SYMMETRY PLANE. (4) a, with three common symmetry planes making right angles with each other. (5) b, with one common symmetry plane. (6) c, with no symmetry plane what- ever. Isometric System. Tetragonal System. Hexagonal System. Orthorhombic System. Monoclinic System. Triclinic System. USE OF CRYSTAL AXES. In addition to the symmetrical arrangement of the faces it has been noticed that, however numerous may be the faces on a crystal, they bear further definite relationships to each other in that, within certain limits, the facial angles appear to be fixed. The determination of these facial angles 6 is an important part of the study of crystallography, and for this purpose it has been found desirable to refer all the faces of a crystal to three (sometimes four) arbitrarily chosen fixed lines. If we can determine the inclinations which the different faces make to these fixed lines we can thereby determine their mutual inclinations to each other. These fixed lines are called Crystal Axes. Theoretically these crystal axes might be chosen mak- ing any desired angles to each other, provided the angles be known, but for the sake of simplicity it is found desirable to choose them so that they may bear some definite relation- ship to the symmetry of the crystal. Therefore, we have the following GENERAL RULE FOR CHOOSING CRYSTAL AXES. In all the six systems the crystal axes are chosen to coincide with symmetry axes, as far as this is possible ; a principal symmetry axis always being given the preference. PARAMETERS. All the crystal axes are supposed to meet and cross each other at the geometric centre of the crystal. In ascertain- ing the inclinations of the faces to each other by means of the crystal axes we conceive both faces and axes to be ex- tended indefinitely, or until the plane of any face under con- sideration cuts all three axes, or until it is evident that the plane never will cut the axis. The relative distances at which the plane of a face cuts the three axes, measured from the centre of the axial cross, are called the parameters of the face. Experience has shown that the value of a parameter is either infinity or some small rational quantity. The parameter value infinity is expressed thus, oo, and in- dicates that the face is parallel to the axis. CRYSTAL FORM. In studying crystals in any one system it will be noticed that, if we start with any face with known parameter values, in order that the law of symmetry of the system may 7 be fulfilled there must also occur a definite number of other faces with exactly the same parameter values as has the face we start with. In speaking of crystal form, then, we use the term in a technical sense and mean all the faces taken together that are required to complete the symmetry of the system. Crystal form, therefore, may be defined as the sum of all those crystal faces, which the symmetry of a crystal demands, when one of them is present. LAW OF SYMMETRY. Both ends of either a symmetry or a crystal axis, and the ends of all interchangeable axes must be cut by the same num- ber of crystal faces, similarly placed. This is a fundamentally important law, and holds good with certain modifications to be noted beyond. ISOMETRIC SYSTEM. This system has three principal symmetry planes at right angles to each other ; also six common symmetry planes lying intermediate between and diagonal to the prin- cipal symmetry planes, and making with each other angles of 60, 90 and 120. The three principal symmetry planes divide the crystal body into eight solid parts called octants. The three principal symmetry axes are interchange- able and, in accordance with the rule, are chosen as the three crystal axes. The law of symmetry, therefore, in this system demands that the six ends of the three interchange- able axes should be cut by the same number of planes, similarly placed, i. e., the six ends of the crystal axes must come out in similarly located points. A little thought will also show that all eight octants must contain the same number of crystal faces. Crystals in this system show an equal development in three directions. 8 CRYSTAL FORMS IN THE ISOMETRIC SYSTEM. In determining crystal forms it will suffice if we can determine for a form how any one of its faces, if extended, would cut the three crystal axes, inasmuch as all the other faces of the same form must cut the axes at the same dis- tances. We can indicate how the plane of a face cuts the three axes by means of the three parameter values expressed in the form of a ratio. Thus, a : nb : me, where the letters a, b and c represent the three crystal axes, and the coeffi- cients m and n give the parameter values. In the isometric system, as all three axes are interchangeable, they must all be treated alike, and it is not necessary to distinguish be- tween a, b and c. We will designate all three, therefore, by the same letter, a. The parameter values expressed in the form of a ratio may be called the SYMBOL of a crystal face, and also of the form to which the face belongs. In the table given below will be found all the possible variations of crystal symbols in the isometric system, so ar- ranged as to make it quite evident that no variations are left out. ISOMETRIC HOLOHEDRAL FORMS. Three axes cut alike. __ a : a : a Octahedron 8 Faces /// ^ Two axes cut alike Two axes cut at distance j^the other __ __ a a : a : : a : ma oca Trisoctahedron Dodecahedron 24 Faces * h ' '' 12 Faces / o i Two axes cut at distance*^ the other ^ a a : ma : : oca : ma oca Trapezohedron Hexahedron 24 Faces /?// 6 Faces / 00 Three axes cut unlike _ a. a : ma : : ma : na cca Hexoctahedron Tetrahexahedron 48 Faces ' 7 / ' j> JT 7 I 1 J 7 four alternating octants \obiained by dividing the holohedral form by the three principal symmetry planes], while the remain- ing faces are developed. By this process we suppress faces on one side of a principal symmetry plane and develop on the other side. This necessarily destroys the principal symmetry plane. Inclined hemihedral forms, therefore, may be distin- guished by the fact that they have the six common symmetry planes, but not the principal symmetry planes of the isometric system. The following forms result from the application of the above law. Octahedron gives Tetrahedron with 4 faces Trapezohedron gives Trigonal tf igtetrahed^on with 12 faces Trisoctahedron gives Tetragonal S8B68(18HlSSL _ _ with 12 faces Hexoctahedron gives Hextetrahedron with 24 faces Hexahedron gives Hexahedron with 6 faces Dodecahedron gives Dodecahedron with 12 faces Tetrahexahedron... gives Tetrahexahedron with 24 faces The first four of the above forms occur with just half as many faces as have the corresponding holohedral forms, but the last three occur with all the faces present in the holohedral forms. They may appear to be holohedral, but are really hemihedral forms. Hemihedral forms, there- fore, cannot always be distinguished from holohedral by the number of their faces. The reason why these three forms occur with all the faces of the corresponding holohedral forms is evident when we consider that they have infinity in their symbol, i. e., each face, being parallel to an axis, must lie in two adjacent octants and cannot, therefore, be suppressed according to the rule for inclined hemihedral forms. II. PARALLEL HEMIHEDRAL FORMS IN THE ISOMETRIC SYSTEM. Parallel hemihedral forms may be conceived to be devel- oped by suppressing on each of the seven holohedral forms all -12- the faces that lie wholly within twelve alternating parts ob- tained by dividing the holohedral form by means of the com- mon symmetry planes, while the remaining faces are developed. By this process we destroy planes on one side of a common symmetry plane and develop those on the other side. This necessarily destroys the common symmetry planes. Parallel hemihedral forms, therefore, may be distin- guised by the fact that they have the three principal symmetry planes but not the six common symmetry planes of the isomet- ric system. The following forms result from the application of the above law : Tetrahexahedron gives Pentagonal dodecahedron __ with 12 faces Hexoctahedron gives Didodecahedron with 24 faces Octahedron .gives Octahedron with 8 faces Hexahedron gives Hexahedron with 6 faces Dodecahedron gives Dodecahedron with 12 faces Trapezohedron gives Trapezohedron with 24 faces Trisoctahedron gives Trisoctahedron with 24 faces In this case there are only two forms that occur with half as many faces as do the corresponding holohedral forms. The last five forms, though apparently holohedral, are really hemihedral. . The reason why five forms occur with as many faces in the parallel hemihedral division as in holohedral is similar to that given for the inclined hemihedral forms. Each face is so placed that it lies in at least two adjacent parts, and therefore cannot be suppressed. III. 6YROIDAL HEMIHEDRAL FORMS IN THE ISOMETRIC SYSTEM Gyroidal hemihedral forms may be conceived to be devel- oped by suppressing on each of the seven hoi oliedral forms the faces lying wholly within twenty-four alternating parts obtained by cutting the Jwlohedral forms into forty-eight sections by means of botli sets of symmetry planes, while the remain- ing faces are developed, By this process all the symmetry planes are destroyed, and gyroidal hemihedral forms may be distinguished by this tact. 13 As there is but one holohedral form with forty-eight faces, this can evidently be the only form whose faces lie wholly within the forty-eight parts into which the nine sym- metry planes cut a crystal. Therefore, the hexoctahedron can be the only form giving a new hemihedral form. This is called the pentagonal icositetrahedron. There are two of these, distinguished as right and left-handed, differing from each other only as a right-handed glove differs from a left- handed. Gyroidal hemihedral forms are very rare and are of no importance from a practical point of view. TETARTOHEDRAL FORMS IN THE ISOMETRIC SYSTEM. Tetartohedral forms are also very rare in the isometric system. As a description of these forms is not thought to be in accord with the object of these notes, the reader is referred to larger treatises on crystallography and mineral- ogy, The principles upon which tetartohedral forms are conceived to be developed will be found set forth in these Lecture Notes under the hexagonal system. GENERAL REMARKS ON HEMIHEDRAL AND TETARTOHEDRAL FORMS. No substance crystallizes both holohedral and hemi- hedral or tetartohedral, and two kinds of hemihedral forms, or hemihedral and tetartohedral forms are never found on the same crystal. We do find, however, apparantly holohe- dral, (but really hemihedral forms) occurring with other hem- ihedral forms. The symbols of hemihedral forms are the same as those of holohedral forms, but they are written in the form of a fraction with 2 for a denominator. Similarly, tetartohedral forms have the denominator 4. Holohedral forms really give two hemihedral forms which are usually exactly alike, except in position, and are designated as positive and negative. Thus, the octahedron gives a + tetrahedron and a tetrahedron. The same holds true for all the systems. 14 TABLE GIVING THE SYMBOLS OF THE FORMS IN THE ISO- METRIC SYSTEM AS USED BY WEISS, NAUMANN, DANA AND MILLER. WEISS NAUMANN. DANA. MILLER. Octahedron B a a, o I (Ill) Hexahedron a ooa on a CO O CO H (100) Dodecahedron _ __ a a on a ooO i (110) Trisoctahedron . . a a ma mO m (hhl) Trapezohedron a ma ma m O m m-m Ml) Tetrahexahedron a ma on a co O n i-n (hkO) Hexoctahedron a ma na m O n m-n (hkl) TETRAGONAL SYSTEM. In this system there is but one principal symmetry plane and four common symmetry planes all at right angles to the principal symmetry plane. These four common symmetry planes occur as two pairs of right angled planes, each pair standing intermediate between or 45 inclined to the other pair. There is, therefore, one direction, namely, that of the principal symmetry axis independent of and distinguished from the others. The two common symmetry axes of each pair are interchangeable with each other, but not with the principal symmetry axis, nor with the common symmetry axes of the other pair. In choosing the three crystal axes we will, in accordance with the general ride, select the. principal symmetry axis for one crystal axis, and for the two other crystal axes we will select either one of the two pairs of common symmetry axes. In studying the crystal forms it is customary to place the principal symmetry plane horizontal. The principal symmetry axis, therefore, becomes the vertical crystal axis. The two other selected crystal axes are horizontal, one of them being placed from front to back. The two horizontal axes, being interchangeable, are both designated by the let- ter a, and the vertical axis by the letter c. The vertical axis c can never be equal to the horizontal axis a, and the ratio between c and a can never be a rational -15- + V*v jJ 4 quantity. In the symbols given below, however, the c"6^ : efficients before c and a are always rational quantities and may also be equal. As ic can never be equal to la, m may become equal without changing the character of the form. to HOLOHEDRAL TETRAGONAL FORMS. a a me Direct pyramid 8 Faces First Order a a OOC Direct prism 4 Faces a a na na me ccc Ditetragonal pyramid Ditetragonal prism___ 16 Faces 8 Faces a ooa me Indirect pyramid. 8 Faces Second Order. _ a ooa occ Indirect prism __ 4 Faces oca ooa c Basal pinacoid 2 Faces COMBINATION OF FORMS. Only in case of one of the three pyramids can a single crystal form entirely bound a crystal. The other forms can occur only in com- bination. The basal pinacoid and the prisms of the first and second orders have no variable in their symbol. They can occur therefore but once on a crystal. All the other forms are variable and can occur an indefinite number of times on the same crystal. The forms in this system can be readily determined without the aid of the symbols by means of the following RULES FOR DETERMINING TETRAGONAL FORMS. First. A face which is parallel to both horizontal axes is the basal pinacoid. Second. A face whose plane cuts all three axes be- longs to a pyramid. If it cuts the two horizontal axes alike it belongs to a direct pyramid; if unlike, to a ditetragonal pyramid. Third. A face whose plane cuts the vertical axis and is parallel to one of the horizontal axes belongs to an in- direct pyramid. ff * T-^6^^-^10 & yt Lower faces if # -8- 4 7^ X ~* S^Uftt 12 Leaving faces 2, 6, 10 above, and 4, 8, 12 below to be developed. Third. Suppression through pyramidal and trapezohe- dral hemihedrisms. Upper faces ^ f\ 4 ^ 6 ^'8 \ 10 *t 12 Lower faces -i- _\-3-\-6-X-^X-e-^tt H, Leaving faces 2, 4, 6, 8, 10, 12 above, and none below. It is evident that the last case does not give a form that fulfills the above conditions, but these conditions are found to be fulfilled in the first two cases. We have, there- fore, two possible kinds of tetartohedrism. First. The tra- pezohedral tetartohedrism, developed by the simultaneous application of the rhombohedral and the trapezohedral hemihedrisms, and second, the rhombohedral tetartohedrism, developed by the simultaneous application of the rhombo- hedral and the pyramidal hemihedrisms. I. TRAPEZOHEDRAL TETARTOHEDRISM. If this method be applied to the seven holohedral forms it is found that two of them can give no new form, as none of their faces can be suppressed. They are the basal pina- coid and the prism of the first order. 24: Basal pinacoid gives Basal pinacoid. Pyramid of the first order gives Rhombohedron. Prism of the first order gives Prism of the first order. Pyramid of the second order gives Trigonal pyramid. Prism of the second order gives Trigonal Prism. Dihexagonal pyramid gives Trigonal trapezohedron. Dihexagonal prism gives Ditrigonal prism. The tetartohedral rhombohedron cannot be distinguished from the hemihedral rhombohedron. It occurs both posi- tive and negative. The trigonal prism has three faces that would give in cross-section an equilateral triangle. The trigonal pyramid has three faces above lying ex- actly over the three faces below. It is symmetrically placed in reference to the axes. The trigonal trapezohedron has three faces above that do not lie exactly over the three faces below, nor do they occur in alternating position as is the case with the rhom- bohedron. They may be recognized by the unsymmetrical position of each face with reference to the axes, or with reference to the prism, which is usually present. The ditrigonal prism has six faces that make alternately sharper and blunter vertical edges. In case the trigonal trapezohedron is present a crystal may be recognized as tetartohedral by the fact that no symmetry plane whatever is present. II. RHOMBOHEDRAL TETARTOHEDRISM. The forms in this division are by no means so impor- tant as are those in the foregoing division, and are repre- sented in nature by only a few not very common minerals. A very brief summary of the possible forms, therefore, will suffice. Basal pinacoid gives .Basal pinacoid. Pyramid of first order gives. -Rhombohedron of first order. Prism of first order gives Prism of first order. Pyramid of second order gives Rhombohedron of second order. Prism of second order gives Prism of second order. Dihexagonal pyramid gives Rhombohedron of third order. Dihexagonal prism gives Prism of third order. 25 The prism of the third order differs from the prisms of the first and second orders only in its position, it being un- sym metrical with reference to the crystal axes. Similarly, the third order rhombohedron has an unsymmetrical posi- tion, but otherwise does not differ from the other rhom- bohedrons. The symbols of tetartohedral forms may be expressed in the form of a fraction with the usual holohedral symbols for numerators and 4 for the denominator. For a further description of the symbols as well as for fuller descriptions of these tetartohedral forms the reader is referred to more extended treatises on crystallography.* HEMIMORPHIC HEXAGONAL FORMS. Hemimorphic forms are half-forms that differ from hemihedral forms in a very important respect, namely, the opposite ends of some (holohedral) symmetry axis are cut by different planes. Usually some form or forms that are present on one end are wanting at the other end of the axis. In the hexagonal system hemimorphic forms play a very important and interesting role in the case of a very common mineral, tourmaline. Here we have the hemi- morphism superimposed upon a rhombohedral hemihe- drism. This gives a sort of quarter-form. The forms that commonly occur hemimorphic in this mineral, i. e., differ- ently developed at the two ends of the vertical axis, are rhombohedrons, scalenohedrons and the basal pinacoid. In addition to these there are usually to be seen a trigonal prism of the first order, a ditrigonal prism and the prism of the second order. These forms differ from tetartohedral forms in that we have a trigonal prism in combination with a scalenohedron, and by the fact that the trigonal prism is of the first order, instead of the second order. The trigonal prism, therefore, * See Elements of Crystallography, by George H. Williams, New York, Henry Holt & Co., 1890. 26 lies under the rhombohedron, i. e., makes a horizontal edge with the rhombohedron. The explanation of the trigonal prism is as follows : The hexagonal prism of the first order may be considered to be the equivalent of a rhombohedron with infinite c axis, three faces belonging to the upper and three to the lower half of the crystal. If, now, the crystal is hemimorphic, the three prism faces that belong to one end of the crystal may be developed while the three belonging to the other end may be suppressed. In the same way the ditrigonal prism may in this case be considered to be the hemimorphic form of a scalenohedron with infinite c axis. TABLE OF SYMBOLS IN THE HEXAGONAL SYSTEM. WBIS9. NAD- DANA. MILLER- MANN. BRAVAIS. First order pyramid . a a ooa : me mP m (hohi) Second order pyramid 2a a 2a : me mP2 m-2 (hh2h2i) Dihcxagonal pyramid pa a na : me mPn m-n (hkli) First order prism a a ooa : ooc OOP I (10lO) Second order prism . . 2a a 2a : ooc coP2 i-2 (1020) Dihexagonal prism. . pa a na : coc oo Pn i-n (khlb) Basal pinacoid. ooa ooa ooa : c OP O (0001) ORTHORHOMBIC SYSTEM. This system belongs to the class with no principal symmetry plane. It has, however, three common symmetry planes at right angles to each other. There are, there- fore, no interchangeable symmetry axes. In accordance with the rule for the selection of crystal axes the three common symmetry axes become the crystal axes. None of these axes is pre-eminent above the others, as there is no principal symmetry axis. We may select, then, any one of the three axes for the vertical axis, the two others becoming the horizontal axes. The shorter of the horizontal axes is placed from front to back and is called 27 the brachy (short) axis. It is designated by the letter a. The longer horizontal axis runs, therefore, from right to left, and is called the macro (long) axis. It is designated by the letter b. The vertical axis is designated by the let- ter c. In this and the following systems the unit values for the three axes are different, and the ratio between them is an irrational one. On the other hand the co-efficient values are always simple rational quantities. CRYSTAL FORMS IN THE ORTHORHOMBIG SYSTEM. There, are three kinds of forms in this system. First. Forms with eight faces Pyramids. Second. Forms with four faces Prism and Domes. Third. Forms with two faces.. __Pinacoids. Pyramid na Prism na Macro-dome a Brachy-dome GO a Macro-pinacoid U Brachy-pinacoid oca b b cob b cob b GO b me coc me me coc coc c Basal pinacoid ooa Pyramids cut all three axes. Prisms and domes cut two axes and are parallel to one. Pinacoids cut but one and are parallel to two axes. The relative lengths of any two axes cannot be determined by the thickness of a crystal, but only by means of some face which, if extended, would cut the two axes in question. E.g., the prism face which cuts the two horizontal axes determines which is the longer and which is the shorter. The dome determines the relative lengths of the vertical and one of the horizontal axes. The pyramid face, which cuts all three axes determ- ines the relative lengths of the three axes. It is evident that the prism and the domes are virtually the same thing with different names, as we can change either dome into a prism by a different selection of the ver- tical axis. In a certain sense, therefore, the domes and the prism are interchangeable. In the same sense the three pinacoids are also interchangeable. 28 RULES FOR DETERMINING FORMS IN THE ORTHORHOMBIC SYSTEM. First. A face whose plane cuts all three axes belongs to a pyramid. Second. A face parallel to one axis belongs either to a prism or to a dome ; if parallel to the vertical axis it belongs to a prism ; if parallel to the brachy (short) axis, to a brachy-dome; if parallel to the macro (long) axis, to a macro-dome. Third. A face parallel to two axes belongs to a pina- coid ; if parallel to the vertical and brachy axes it belongs to the brachy-pinacoid ; if parallel to the vertical and macro axes, to the macro-pinacoid ; if parallel to the two horizontal axes, to the basal pinacoid. Fourth. There may occur on the same crystal an in- definite number of pyramids or of prisms or domes, because these forms have variable symbols ; but the three pinacoids can occur but once, as their symbols are invariable. HEMIHEDRAL AND HEMIMORPHIG FORMS IN THE ORTHORHOMBIC SYSTEM. There is but one kind of hemihedrism possible in this system, namely, the one corresponding to the inclined in the isometric system, and to the sphenoidal in the tetra- gonal system. This is produced by cutting the holohedral forms by means of the common symmetry planes into octants, and by suppressing the faces that lie wholly within alternating octants while the remaining faces are developed. There is only one form that can produce a new form, and this is the pyramid. This produces the orthorhombic sphenoid, which is similar to the tetragonal sphenoid, but without interchangeable axes. Hemimorphic forms are fairly common in this system. They are developed by suppressing certain forms at one end of a symmetry axis that are developed at the other end. 29 TABLE OF SYMBOLS IN THE ORTHORHOMBIC SYSTEM. WEISS. NAUMANN. DANA. MILLER. Pyramid na b me mPn m-n (hkl) Prism na b ooc mP I (110) Brachy-dome Macro-dome ooa a b QO b me me oo Pn oo Pn i- i-n (khO) h > k (hkO) h > k Brachy-pinacoid . Macro-pinacoid . . Basal pinacoid ooa a ooa b cob QO b ooc OOC c ooP& QOPOO OP i.r i-T O (010) (100) (001) MONOCLINIC SYSTEM. In this system there is but one symmetry plane and, therefore, but one symmetry axis. In accordance with the rule we select this symmetry axis for one of the crystal axes. It is the only natural crystal axis. The two remaining axes are selected to lie in the symmetry plane and parallel to prominent edges, or, if no edge is available, parallel to a prominent face. (Only in rare exceptions must a line connecting two corners be chosen). If an axis be parallel to an edge it must neces- sarily be parallel to the faces forming the edge. Now, as edges parallel to the symmetry plane in the monoclinic system are not found to occur exactly at right angles to each other, the two axes in the symmetry plane are also never at right angles. We have, then, two axes lying in the symmetry plane oblique to each other, and one axis coinciding with the symmetry axis, and, therefore, at right angles to the other two axes. In orienting the crystal the symmetry plane is placed vertical and from front to back, so that the symmetry axis becomes the b axis, and is called the ortho-axis. The crystal is then turned until one of the oblique axes becomes vertical and the other slopes downward from the center of Of) Ov the crystal towards the observer, i, e., towards the front. The vertical axis then becomes the c axis ; and the other, or clino-axis the a axis. CRYSTAL FORMS IN THE MONOCLINIC SYSTEM. First. Forms with four faces prisms, clino-domes and pyramids. Second. Forms with two faces ortho-domes and pinacoids. f_ (Positive.. _ -na First I Pyramids J Nega tive '_"_"" +na i Prism na ^Clino-dome ooa Positive -a Negative +a Second. J Ortho-pinacoid a I Basal pinacoid ooa ^Clino-pinacoid oca Ortho-domes b b b b oob oob oob QO b b me me ooc me me me ooc c QOC The plane in which lie the vertical and the ortho-axes divides the crystal into two unsymmetrical parts. There- fore, the faces that may occur on one side of this plane are not repeated on the other side, except that every face must have an opposite parallel face. For instance, a face in the pyramid position cutting the vertical and ortho-axes and the front end of the clino axis need not occur at the rear of the crystal in a symmetrical position. The two pyramid faces in front, above the basal pinacoid, together with the two opposite and parallel faces at the back, below the basal pinacoid, make up a monoclinic pyramid which is called a partial pyramid. It takes two of these partial pyramids to correspond with the orthorhombic pyramid. For a similar reason there are two partial ortho- domes. In the case of these partial forms if a face cuts the ends of the vertical and clino axes that form an obtuse angle it belongs to a negative partial form. If it cuts the ends of these axes forming the acute angle the form is positive. The pyramid, prism and clino-dome are virtually the same thing, inasmuch as they can be changed the one into the other by changing the locations of the two oblique axes. 31 For the same reason the basal pinacoid, ortho-pinacoid and ortho-dome may be changed the one into the other. The clino-pinacoid is the only form that cannot be changed into an other by changing the position of the oblique axes, for the reason that it is the only form parallel to a symmetry plane. RULES FOR DETERMINING FORMS IN THE MONOCLINIG SYSTEM. First. A face whose plane cuts all three axes belongs to a pyramid. If it lies over the acute angle of the oblique axes it belongs to a positive pyramid ; if over the abtuse angle, to a negative pyramid. Second. A face parallel only the vertical axis belongs to a prism. Third. A face parallel only the clino-axis belongs to a clino-dome. Fourth. A face parallel only to the ortho-axis belongs to an ortho-dome. Fifth. A face parallel to two axes is a pinacoid ; if it is parallel to the clino and the ortho-axes it belongs to the basal pinacoid ; if to the vertical and the clino-axes, to the clino-pinacoid ; if parallel to the vertical and the ortho- axes, to the ortho-pinacoid. Sixth. All the forms except the three pinacoids may occur an indefinite number of times on the same crystal because they have variable symbols. HEMIHEDRAL AND TETARTOHEDRAL FORMS IN THE MONOCLINIG SYSTEM. Tetartohedral forms are not certainly known to occur. Hemihedral forms are known to exist, but their occurrence is so rare that a discussion of these forms is considered beyond the scope of these lecture notes. 32 TABLE OF SYMBOLS IN THE MONOCLINIC SYSTEM. WKJSS. NAUMANN. DANA. MILLER Pyramid j Positive.. ( Negative... Prism ._ -na +na , na oca b b b b me me DOC me + mPn -mPn ocP mP^o + m-n -m-n I m-> (hkl) (hkl) (110) (Okl) Clino-dome Ortho-dome j P sitive ( Negative Orthopioacoid .. . -a +a a ooa oob GO b oob oo b me me ooc c + mPoB -mPo> ooPoo OP + m-7 -m-T i-i O (hOl) (hOl) (100) (001) Basal pinacoid Clino pinacoid.. QO a b COC GO POO i x i (010) TRICLINIC SYSTEM. In this system, as there is no symmetry plane, and, therefore, no symmetry axis, there can also be no natural crystal axes. In selecting the crystal axes, it is customary to chose directions parallel to prominent edges on the crystal, seeking at the same time to secure axes as near right angles as the crystal will allow. (Sometimes it will be necessary to chose a direction parallel to two faces that do not meet in an edge, or a diagonal through the crystal connecting two corners.) As edges and faces in the triclinic system are never exactly at right angles, although approximately so, we can- not have crystal axes at right angles. As is the case in the orthorhombic system, we choose any one of the three axes for the vertical axis, place the longer of the other two from right to left and the third, therefore, as nearly from front to back as the obliquity of the axes will allow. The shorter axis, a, is called the brachy-axis ; the longer, b, the macro-axis, exactly as in the orthorhombic system. Each form in this system consists of but pairs of parallel planes. There is no essential difference between 33 them, as any form may be changed into any other form merely by making a different selection of the crystal axes. Forms are named as they are in the orthorhombic system. RULES FOR DETERMINING CRYSTAL FORMS IN THE TRIGLINIO SYSTEM. First. All faces whose planes cut all three axes belong to pyramids. There are four cf these partial pyramids to two in the monoclinic and to one in the orthorhombic systems. Second. Faces parallel only to the vertical axis belong to prisms. There are two of these partial prisms. Third. Faces parallel only to the brachy-axis belong to brachy-domes. There are two of these partial brachy - domes. Fourth. Faces parallel only to the macro-axis belong to macro- domes. There are two of these partial macro- domes. Fifth. Faces parallel to the brachy and macro-axes belong to the basal pinacoid. Sixth. Faces parallel to the vertical and brachy-axes belong to the brachy-pinacoid, Seventh. Faces parallel to the vertical and macro- axes belong to the macro-pinacoid. Eighth. The pyramids, prisms, brachy and macro- domes, having variable symbols, may occur an indefinite number of times on a crystal ; all the other forms but once. SYMBOLS. The symbols used are exactly the same as those given for the orthorhombic system. To distinguish between the different partial forms special accents and positive and negative signs are used. 34 DISTORTION OF CRYSTALS. In all the foregoing cases we have had under considera- tion ideally developed crystals or crystal models. In nature, however, perfect crystals are very rare. Almost in- variably they are to some extent distorted. Except in the case of mechanical distortion this crystal distortion is usually of such a nature that the inclinations of the faces to each other and to the axes remain unaltered. This may be conceived to be accomplished by the shoving of one or more faces parallel to themselves so that some faces are in- ordinately developed at the expense of other faces. In this way some faces may be completely crowded off the crystal with an apparent loss of symmetry. In natural crystals, therefore, we cannot recognize symmetry planes by the symmetrical development of faces on both sides of a plane, but merely by the equal inclina- tions of corresponding faces to the symmetry plane. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. 1940 APR B I940 MAY '940 MAY i9 194 M^l/ 1 *-t HQAH M i LD 21-100m-7, '39(402 / crystallc THE UNIVERSITY OF CALIFORNIA LIBRARY ^{sffi^aiyffiCTSBj mm