Week 2 Compilation PDF

 Gypsum Monoclinic Dolomite Triclinic Crystallography Emerald Hexagonal Andalusite Orthorhombic Garnet Isometric ...

 Gypsum Monoclinic Dolomite Triclinic Crystallography Emerald Hexagonal Andalusite Orthorhombic Garnet Isometric  Symmetry An ordered arrangement of atoms imparts crystals and minerals the property of symmetry. Perfectly formed crystals show a symmetrical arrangement of crystal faces since the location of the faces is controlled by the arrangement of atoms in the crystal structure. , Beryl var. Aquamarine, Erongo Mountains, Namibia  AXIS OF SYMMETRY Axis of symmetry: It is an imaginary straight line that passes through the center of a crystal, when the crystal rotates around this axis similar face, edge or solid angle appear. Rotation: It is a process of several appearances of similar faces or edges or solid angles when the crystal is rotated 3600 degrees. Axes of symmetry  AXES OF SYMMETRY Axis of Rotation = imaginary line through a crystal about which the crystal may be rotated and repeat itself in appearance (1,2,3,4 or 6 times during a complete rotation FOUR FOLD ROTATION AXIS THREE FOLD ROTATION AXIS  AXIS OF SYMMETRY 1-Fold Rotation Axis - An object that requires rotation of a full 360o in order to restore it to its original appearance has no rotational symmetry. Since it repeats itself 1 time every 360o it is said to have a 1-fold axis of rotational symmetry  2 Fold axis of symmetry 2-Fold Rotation Axis - If an object appears identical after a rotation of 180o, that is twice in a 360o rotation, (3600/1800 = 2). then it is said to have a 2-fold rotation axis (3600/180 0= 2).  3 Fold axis of rotation 3-Fold Rotation Axis- Objects that repeat themselves upon rotation of 120o are said to have a 3-fold axis of rotational symmetry (3600/1200 =3), and they will repeat 3 times in a (3600/1200) 360o rotation. A filled triangle is used to =3 symbolize the location of 3-fold rotation axis.  4 Fold axis of rotation 4-Fold Rotation Axis - If an object repeats itself after 90o of rotation, it will repeat 4 times in a 360o rotation, as (3600/900= illustrated previously. 4) A filled square is used to symbolize the location of 4-fold axis of rotational symmetry.  6 Fold axis of symmetry 6-Fold Rotation Axis - If rotation of 60o about an axis causes the object to repeat itself, then it has 6-fold axis of (3600/600=6). rotational symmetry (3600/600=6). A filled hexagon is used as the symbol for a 6-fold rotation axis.  Axis of rotation Although objects themselves may appear to have 5-fold, 7- fold, 8-fold, or higher-fold rotation axes, these are not possible in crystals. The reason is that the external shape of a crystal is based on a geometric arrangement of atoms. Note that if we try to combine objects with 5-fold and 8-fold apparent symmetry, that we cannot combine them in such a way that they completely fill space, as illustrated below. There are the following types of axes of symmetry: a) Daid axis of symmetry. , Repetition takes place every 180 degrees b) Triad axis of symmetry. Repetition takes place every 120 degrees c)- Tetrad axis of symmetry. Repetition takes place every 90 degrees d)- Hexad axis of symmetry.  2-fold axis of symmetry During a 360° rotation of the box around the axis, the exact same image is shown  2-fold axis of symmetry (diagonal axis of symmetry) Again during a 360° rotation of the box around the axis, the exact same image is shown twice.  3A22-fold axis of symmetry hree axis of 2 fold rotation And again, during a 360° rotation of the box around the axis, the exact same image is shown  SYMMETRY OF ELEMENTS Planes of symmetry Planes of symmetry can be regarded as mirror planes. They divide a crystal in two. Each side of the division is the mirror of the other while the total image is not altered by the mirror plane (the symmetry stays intact). Symmetry Symmetry No symmetry present  Plane of Symmetry Mirror Plane = imaginary plane that divides a crystal into halves, each of which is the mirror image of the other  Plane of Symmetry First plane of Second plane Third plane of symmetry of symmetry symmetry 3m In all the above images, the dividing plane acts as a mirror plane  Symmetry Symbols Reflection (m)- produced by a mirror plane that passes through a crystal structure so the pattern on one side is a mirror image of the pattern on the other. (symmetry over a plane) How many mirror planes do these crystals have? Both have 3! So notation is 3m Center of symmetry (i): A crystal has a center of symmetry when like faces, edge or solid angles are arranged in pairs in corresponding positions and on opposite sides of the center point. No Yes No No Yes center center center center center  Center of Symmetry Center of symmetry (i)  The central point from which crystal faces and edges appear the same on either end of the center  ROTOINVERSION Combinations of rotation with a centre of symmetry Objects that have rotoinversion symmetry have an element of symmetry called a rotoinversion axis.  ROTOINVERTION  A 1-fold rotoinversion axis is the same as a centre of symmetry  ROTOINVERSION 2-fold Rotoinversion - The operation of 2-fold rotoinversion involves first rotating the object by 180o then inverting it through an inversion centre.  This operation is equivalent to having a mirror plane perpendicular to the 2-fold rotoinversion axis.  ROTOINVERSION 3-fold Rotoinversion - This involves rotating the object by 120o (360/3 = 120), and inverting through a centre. (example, cube).  A 3-fold rotoinversion axis is denoted as ( ̅3), (pronounced "bar 3").  ROTOINVERSION 4-fold Rotoinversion This involves rotation of the object by 90o then inverting through a centre. A four fold rotoinversion axis is symbolized as ( ̅4).  ROTOINVERSION 6-fold Rotoinversion A 6-fold rotoinversion axis ( ̅6 ) involves rotating the object by 60o and inverting through a centre  Hermann-Mauguin (International) Symbols The rectangular block shown here has : 3 A 2-fold rotation  3 mirror planes (m)  center of symmetry (i).  Hermann-Mauguin (International) Symbols 3 A2 FOLD ROTATIO N Write a number representing each of the unique rotation axes present 2 2 2. A unique rotation axis is one that exists by itself and is not produced by another symmetry  Hermann-Mauguin (International) Symbols 3 A2 FOLD ROTATION Next we write an "m" for each unique mirror plane. 2 2 2 2m 2m 2m a unique mirror plane is one that is not produced by any other symmetry operation  Hermann-Mauguin (International) Symbols 3 A2 FOLD ROTATION If any of the axes are perpendicular to a mirror plane we put a slash (/) between the symbol for the axis and the symbol for the mirror plane: 2 2 2 2/m 2/m 2/m  Hermann-Mauguin (International) Symbols This model has one 2-fold axis and 2 mirror planes. For the 2-fold axis, we write: 2 RHOMBIC DIPYRAMID  Hermann-Mauguin (International) Symbols Each of the mirror planes is unique. We can tell that because each one cuts a different looking face. So, we write 2 "m"s, one for each mirror plane: 2m m or RHOMBIC DIPYRAMID Crystal Form: It is a group of crystal faces that have similar shape and area and display the same physical and chemical properties.. Seven Forms Two One One Six Forms Forms Form Form Three Forms Type of Crystals There are two types of crystal depending on the number of forms. 1- Simple crystal: The crystal consists of only one form. 2- Combine crystal: The crystal consists of more than one form.  Classification of Form I. Depend on closing and opening of the space: A.Close Form..– if the faces consisting a form enclose a space B. Open form – if the faces do not enclose a space dom e  CRYSTAL FORMS Names of crystal forms: A) Isometric crystal forms B) Non- isometric crystal forms Pyrite  -Isometric Crystal Forms No.of No.of Name Name Faces Faces (1) Cube 9)Tristetrahedron 6 12 (2) Octahedron (10) Hextetrahedron 8 24 (3) Dodecahedron (11) Deltoid dodecahedron 12 24 (4) Tetrahexahedron (12) Gyroid 24 24 (5) Trapezohedron (13) Pyritohedron 24 12 (6) Trisoctahedron (14) Diploid 24 24 (7) Hexoctahedron (15) Tetartoid 48 12 (8) Tetrahedron 4  Non-I sometric Crystal Forms Number Number Name Name of Faces of Faces (16) Pedion* 1 (32) Dihexagonal pyramid 12 (17) Pinacoid** 2 (33) Rhombic dipyramid 8 (18) Dome or Sphenoid 2 (34) Trigonal dipyramid 6 (19) Rhombic prism 4 (35) Ditrigonal dipyramid 12 (20) Trigonal prism 3 (36) Tetragonal dipyramid 8 (21) Ditrigonal prism 6 (37) Ditetragonal dipyramid 16 (22) Tetragonal prism 4 (38) Hexagonal dipyramid 12 (23) Ditetragonal prism 8 (39) Dihexagonal dipyramid 24 (24) Hexagonal prism 6 (40) Trigonal trapezohedron 6 (25) Dihexagonal prism 12 (41) Tetragonal trapezohedron 8 (26) Rhombic pyramid 4 (42) Hexagonal trapezohedron 12 (27) Trigonal pyramid 3 (43)Tetragonal scalenohedron 8 (28)Ditrigonal pyramid 6 (44) Hexagonal scalenohedron 12 (29) Tetragonal pyramid 4 (45) Rhombohedron 6 (30) Ditetragonal pyramid 8 (46) Rhombic disphenoid 4 (31) Hexagonal pyramid 6 (47) Tetragonal disphenoid 4 END OF LECTURE  INTERNAL ORDER IN CRYSTALS AMETHYST QUARTZ Crystalline structures are characterized by a repeating pattern in three dimensions. The periodic nature of the structure can be represented using a lattice. Each block is represented by a point  This array of points is a LATTICE The LATTICE is an ordered arrangement of points in space. MOTIFF consists of the simplest arrangement of atoms which is repeated at every point in the lattice to build up the crystal structure Motiff A crystal can be thought of as being like wallpaper. The motif is analogous to the basis and the arrangement of the motif over the surface is like the lattice. Basis vectors and unit cells b a a and b are the basis vectors for the lattice In 3-D: Unit cell is a small box containing one or more atoms arranged in 3 dimensions. c b UNIT CELL a a, b, and c are the basis vectors for the lattice Lattice parameters- the lengths of the cell edges (a,b and c) and the angles alpha, beta, gamma)  Lattice parameters: c UNIT CELL a b +c b g a +a b g a LATTICE PARAMETERS are the length of the cell +b edges a,b,c, and the angles between them (alpha, beta and gamma) Axial convention: “right-hand rule” A unit cell is defined as a fundamental building block of a crystal structure, which can generate the complete crystal by repeating its own dimensions in various directions. Unit cells repeat to make external crystal form We classify external form in 6 or 7 systems based on the shape of repeating groups of atoms called the unit cell. Unit cells repeat to make external crystal form. Unit cells repeat to make external crystal form  Crystal Systems – Some Definitional information Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal. 7 crystal systems of varying symmetry are known These systems are built by changing the lattice parameters: a, b, and c are the edge lengths , , and  are interaxial angles Fig. 3.4, Callister 7e.  Unit Cells and Crystal Systems There are seven different classes of unit cells that, each defined by different limiting conditions on the unit cell parameters (a, b, c, a, b, g).  CRYSTALLOGRAPHY Crystallographic axes: Imaginary straight lines that intersect at the center of the crystal and extend to the mid of the crystal faces, edges or solid angle.  3-D Lattice Types Name axes angles Triclinic a b c   90o Monoclinic a b c  = 90o 90o Orthorhombic a b c  = 90o Tetragonal a1 = a 2  c  = 90o Hexagonal Hexagonal (4 axes) a1 = a2 = a3  c  = 90o 120o Rhombohedral a1 = a 2 = a 3  90o Isometric a1 = a 2 = a 3  = 90o  Unit Cells Types A unit cell is the smallest component of the crystal that reproduces the whole crystal when stacked together. Primitive (P) unit cells contain only a single lattice point. Internal (I) unit cell contains an atom in the body center. Face (F) unit cell contains atoms in the all faces of the planes composing the cell. Centered (C) unit cell contains atoms centered on the sides of the unit cell. Primitive Face-Centered F P Body-Centered End-Centered I C Combining 7 Crystal Classes (cubic, tetragonal, orthorhombic, hexagonal, monoclinic, triclinic, trigonal) with 4 unit cell types (P, I, F, C) symmetry allows for only 14 types of 3-D lattice.  UNIT CELL Primitive, P Body-centered, I Face-centered, F All of the different possible lattices, including unique centering schemes were determined by A. Bravais and thus the 14 unique lattices are called “Bravais Lattices” Compatible with the orderly arrangement of atoms found in crystals Auguste Bravais  14 Bravais Lattices 14 Types of unit cells that are compatible with the orderly arrangements of atoms found in crystals  c a2 a1 P I Tetragonal a = b = g = 90 o a =a¹c 1 2  c b a P I C F  c c b c P a b I=C R  Bravais Lattices CUBIC LATTICES 3 types; Simple cubic (also called primitive cubic), lattice points only at corners. Body Centered Cubic (BCC), lattice points at corners and in middle of cube. Face Centered Cubic (FCC) lattice points at the corners and in the middle of each face END OF LECTURE  Point Coordinates Coordinates of selected points in the unit cell. The number refers to the distance from the origin in terms of lattice parameters.  PointCoordinates Point Coordinates z 111 Point coordinates for unit cell c center are a/2, b/2, c/2 ½½½ y a 000 b Point coordinates for unit cell x  corner are 111 z 2c  Translation: integer multiple of   lattice constants  identical b y position in another unit cell b 33  Miller Indices, Directions Determine the Miller indices of directions A, B, and C. (c) 2003 Brooks/Cole Publishing / Thomson Learning™ SOLUTION Direction A 1. Two points are 1, 0, 0, and 0, 0, 0 2. 1, 0, 0, -0, 0, 0 = 1, 0, 0 3. No fractions to clear or integers to reduce 4. Direction B 1. Two points are 1, 1, 1 and 0, 0, 0 2. 1, 1, 1, -0, 0, 0 = 1, 1, 1 3. No fractions to clear or integers to reduce 4. Direction C 1. Two points are 0, 0, 1 and 1/2, 1, 0 2. 0, 0, 1 -1/2, 1, 0 = -1/2, -1, 1 3. 2(-1/2, -1, 1) = -1, -2, 2 4. [ 1 2 2]  Crystallographic Directions z Algorithm 1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit cell dimensions a, b, and c y 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas x [uvw] ex: 1, 0, ½ => 2, 0, 1 => [ 201 ] -1, 1, 1 => [ 111 ] where overbar represents a negative index 36 In 1850, Auguste Bravais showed that crystals could be divided into 14 unit cells, which meet the following criteria. The unit cell is the simplest repeating unit in the crystal. Opposite faces of a unit cell are parallel. The edge of the unit cell connects equivalent points. 1850  c a2 a1 P I Tetragonal a = b = g = 90 o a =a¹c 1 2  c b a P I C F  c c b c P a b I=C R  MINERALOGY ROCK MAKING MINERALS I. OLIVINE GROUP ( orthorhombic system) Forsterite MgSiO4(Fo) Fayalite Fe2SiO4(Fa) Forsterite Fayalite  MINERALOGY II. PYROXENE GROUP ( orthorhombic and monoclinic) Orthopyroxene Clinopyroxene Diopside Hypersthene  MINERALOGY III. AMPHIBOLE GROUP Orthorhombic Anthophyllite Series Monoclinic Cummingtonite Series Kupfferite Cummingtonite Cummingtonite Grunerite Grunerite  ROCK FORMING MINERALS AMPHIBOLE GROUP Tremolite Series Tremolite Actinolite Tremolite Actinolit Nephrite e Ferrotremolite Hornblende Series Hornblende Hornblende Lamprobolite Lamprobolite ROCK FORMING MINERALS AMPHIBOLE GROUP  Alkali Amphibole Series Glaucophane Riebeckite Glaucophane Riebeckite  ROCK FORMING MINERALS IV MICA GROUP Muscovite Biotite Lepidolite Biotite Phlogopite Muscovite  ROCK FORMING MINERALS V. SILICA GROUP Quartz Quartz Chalcedony Tridymite Cristobalite Chalcedony  ROCK FORMING MINERALS VI FELDSPAR GROUP Potassium feldspar Plagioclase Series  IGNEOUS PETROLOGY The K-feldspars or alkali felspars: Microcline (Potassium aluminum silicate) Sanidine (Potassium sodium aluminum silicate) Orthoclase (Potassium aluminum silicate Anorthoclase Albite Microli ne  ROCK FORMING MINERALS The plagioclase feldspars: Albite, (Sodium aluminum silicate) Oligoclase, (Sodium calcium aluminum silicate) Andesine, (Sodium calcium aluminum silicate) Labradorite, (Calcium sodium aluminum silicate) Bytownite, (Calcium sodium aluminum silicate) Anorthite, (Calcium aluminum silicate)  IGNEOUS PETROLOGY VII. FELDSPATHOID Group of alkali-aluminum silicates Nepheline Sodalite Leucite Analcite Cancrinite Lazurite Nosean  Other Minerals OXIDES Magnetite Ilmenite Hematite Rutile Corundum Chromite Phosphate Apatite Xenotime Monazite  Sulfides Pyrite Covellite Chalcopyrite Bornite Chalcocite END OF LECTURE Pyroxene minerals  Clinopyroxenes (monoclinic) o Aegirine (Sodium Iron Silicate) o Augite (Calcium Sodium Magnesium Iron Aluminium Silicate) o Clinoenstatite (Magnesium Silicate) o Diopside (Calcium Magnesium Silicate, CaMgSi2O6) o Esseneite (Calcium Iron Aluminium Silicate) o Hedenbergite (Calcium Iron Silicate) o Jadeite (Sodium Aluminium Silicate) o Jervisite (Sodium Calcium Iron Scandium Magnesium Silicate) o Johannsenite (Calcium Manganese Silicate) o Kanoite (Manganese Magnesium Silicate) o Kosmochlor (Sodium Chromium Silicate) o Namansilite (Sodium Manganese Silicate) o Natalyite (Sodium Vanadium Chromium Silicate) o Omphacite (Calcium Sodium Magnesium Iron Aluminium Silicate) o Petedunnite (Calcium Zinc Manganese Iron Magnesium Silicate) o Pigeonite (Calcium Magnesium Iron Silicate) o Spodumene (Lithium Aluminium Silicate)  Orthopyroxenes (orthorhombic) o Hypersthene (Magnesium Iron Silicate) o Donpeacorite, (MgMn)MgSi2O6 o Enstatite, Mg2Si2O6 o Ferrosilite, Fe2Si2O6 o Nchwaningite (Hydrated Manganese Silicate) 1 AMPHIBOLE Orthorhombic series  Anthophyllite (Mg,Fe)7Si8O22(OH)2  Holmquistite Li2Mg3Al2Si8O22(OH)2  Ferrogedrite Fe2+5Al4Si6O22(OH)2 Monoclinic series  Tremolite Ca2Mg5Si8O22(OH)2  Actinolite Ca2(Mg,Fe)5Si8O22(OH)2  Cummingtonite Fe2Mg5Si8O22(OH)2  Grunerite Fe7Si8O22(OH)2  Hornblende Ca2(Mg,Fe,Al)5(Al,Si)8O22(OH)2  Glaucophane Na2(Mg,Fe)3Al2Si8O22(OH)2  Riebeckite (or Crocidolite) Na2Fe2+3Fe3+2Si8O22(OH)2  Arfvedsonite Na3Fe2+4Fe3+Si8O22(OH)2  Richterite Na2Ca(Mg,Fe)5Si8O22(OH)2  Pargasite NaCa2Mg3Fe2+Si6Al3O22(OH)2  Winchite (CaNa)Mg4(Al,Fe3+)Si8O22(OH)2 MICA GROUP Trioctahedral micas Common micas:  Biotite  Lepidolite  Muscovite  Phlogopite  Zinnwaldite Brittle micas:  Clintonite Interlayer deficient micas Very fine-grained micas, which typically show more variation in ion and water content, are informally termed "clay micas". They include 2   Hydro-muscovite with H3O+ along with K in the X site;  Illite with a K deficiency in the X site and correspondingly more Si in the Z site;  Phengite with Mg or Fe2+ substituting for Al in the Y site and a corresponding increase in Si in the Z site. QUARTZ Major varieties of quartz Cryptocrystalline quartz and moganite mixture. The term is generally only used Chalcedony for white or lightly colored material. Otherwise more specific names are used. Agate Multi-colored, banded chalcedony, semi-translucent to translucent Onyx Agate where the bands are straight, parallel and consistent in size. Jasper Opaque cryptocrystalline quartz, typically red to brown Aventurine Translucent chalcedony with small inclusions (usually mica) that shimmer. Tiger's eye Fibrous gold to red-brown colored quartz, exhibiting chatoyancy. Rock crystal Clear, colorless Amethyst Purple, transparent Citrine Yellow to reddish orange to brown, greenish yellow Prasiolite Mint green, transparent Rose quartz Pink, translucent, may display diasterism Rutilated quartz Contains acicular (needles) inclusions of rutile Milky quartz White, translucent to opaque, may display diasterism Smoky quartz Brown to gray, opaque Carnelian Reddish orange chalcedony, translucent Dumortierite Contains large amounts of dumortierite crystals quartz 3  MINERALOGY Study of naturally occurring crystalline substances (minerals) Definition of Terms Crystal – a homogenous solid possessing long-range, three dimensional, internal order. Mineral – a naturally occurring homogenous solid, with definite chemical composition and an ordered atomic arrangement. It is usually formed by inorganic processes. Rock – is an aggregate of minerals. It can be composed of only one kind of mineral (monomineralic) or of different kinds of minerals. Ore Minerals – those minerals from which one or more metals may be extracted at a profit. Industrial Minerals – those minerals which are, themselves, used for one or more industrial purposes such as in the manufacture of electrical and thermal insulators, refractories, ceramics, glass, abrasives, fertilizers, fluxes, cement, and other building materials. Gems – those minerals which have ornamental value, and which possess the qualities of beauty, durability, rarity, fashionability and portability. INTRODUCTION TO CRYSTALLOGRAPHY AND MINERAL CRYSTAL SYSTEMS Steno’s Law – Constancy of Interfacial Angles Nicholas Steno, a Danish physician and natural scientist, found that, by examination of numerous specimens of the same mineral, when measured at the same temperature, the angles between similar crystal faces remain constant regardless of the size or shape of the crystal. The Six Crystal Systems 1. Isometric – (cubic) the three crystallographic axes are all equal in length and intersect at right angles (90o) to each other c a=b=c;α=β=γ=90˚ a b 2. Tetragonal – Three axes, all at right angles, two of which are equal in length (a and b) and one (c) which is different in length. a=b≠c;α=β=γ=90˚ C b a 3. Orthorhombic – Three axes, all at right angles, and all three of different lengths. C a≠b≠c;α=β=γ=90˚ a b 4. Hexagonal – Four axes. Three of the axes fall in the same plane and intersect at the axial cross at 120˚ between the positive ends. These three axes, labeled a 1, a2, and a3 are of the same length. The fourth axis, termed c, may be longer or shorter than the axes set. The c axis also passes through the intersection of the axes set at right angle to the plane formed by the a set. Rhombohedral a=b=c ; α=β=γ=90˚ c a=b≠c;α=β=90˚;γ=120 a3 a2 a1 5. Monoclinic – Three axes, all unequal in length, two of which (a and c) intersect at an oblique angle (not 90°), the third axis (b) is perpendicular to the other two axes. c a≠b≠c; γ≠α=β=90˚ b a 6. Triclinic – The three axes are all equal in length and intersect at three different angles ( any angle but 90°). c a≠b≠c; α≠β≠γ90˚ b a  WHAT IS A MINERAL? A Mineral is a naturally occurring, homogenous solid with a definite, but generally not fixed chemical composition and an ordered atomic arrangement. It is usually formed by inorganic processes. Naturally occurring - formed by a natural process - synthetic products or those produced in the laboratory are not considered minerals example: synthetic ruby is not a mineral Homogenous solid - consists of a single solid substance that cannot be physically subdivided into simpler chemical compounds - excludes gases and liquids - ex. Ice in glacier is a mineral but water is not; liquid mercury fall under mineraloids. Definite chemical composition - means that atoms, or groups of atoms must occur in specific ratios. For ionic crystals (i.e. most minerals) ratios of cations to anions will be constrained by charged balance, however, atoms of similar charge and ionic radius may substitute freely for one another, hence definite, but not fixed.  Not fixed. Ex. Dolomite Ca Mg (CO3)2 is not always pure. It may contain Fe and Mn in place of Mg, therefore chemical formula becomes Ca (Mg,Fe,Mn) (CO3)2.  Ordered atomic arrangement - means crystalline, with three-dimensional periodic arrays of precise geometric arrangement of atoms. -example: both quartz and glass are composed of element Si silicon and O oxygen. The former has a specific arrangement while the latter have random arrangement. Glass lacks consistent atomic order and therefore considered as noncrystalline or amorphous. Another example is graphite and diamond.(Polymorphs-same chem. comp. but have different structures)  Formed by inorganic process - excludes the organically produced compounds - ex. Calcium carbonate in shells, pearl, petroleum and coal - inorganic means pertaining or relating to compound that contains no carbon. The Cubic (Isometric) System Isometric Minerals Crystal Form Example Class Unknown Mineral Listing , , , , Isometric Diploidal Mineral Listing H-M Symbol (2/m 3) Isometric Gyroidal Mineral Listing H-M Symbol (4 3 2) Isometric Hexoctahedral Mineral Listing H-M Symbol (4/m 3 2/m) Isometric Crystallographic Isometric Axes Hextetrahedral Mineral Listing H-M Symbol (4 3m) Isometric Tetartoidal Mineral Listing H-M Symbol (2 3) TETRAGONAL- The Tetragonal System Tetragonal Mineral Crystal Form Example Class Unknown Mineral Listing , , , , Tetragonal Dipyramidal Mineral Listing H-M Symbol (4/m) Tetragonal Disphenoidal Mineral Listing H-M Symbol (4) Tetragonal Ditetragonal Dipyramidal Mineral Listing H-M Symbol (4/m 2/m 2/m) Tetragonal Pyramidal Mineral Listing H-M Symbol (4) Tetragonal Ditetragonal-pyramidal Mineral Listing Tetragonal H-M Symbol (4mm) Crystallographic Axes Tetragonal Scalenohedral Mineral Listing H-M Symbol (4 2m) Tetragonal Trapezohedral Mineral H-M Symbol (4 2 2) The Orthorhombic System Orthorhombic Minerals Crystal Form Class Unknown Mineral Listing. Example , , , , Orthorhombic Dipyramidal Mineral Listing H-M Symbol (2/m 2/m 2/m) Orthorhombic Disphenoidal Mineral Listing Orthorhobic H-M Symbol (2 2 2) Crystallographic Axes Orthorhombic Pyramidal Mineral Listing H-M Symbol (mm2) The Hexagonal System (Hexagonal Division) Hexagonal Minerals Crystal Form Example. , , , , Class Unknown Mineral Listing. Hexagonal Dihexagonal Dipyramidal Mineral Listing H-M Symbol (6/m 2/m 2/m) Hexagonal Dihexagonal Pyramidal Mineral Listing H-M Symbol (6mm) Hexagonal Dipyramidal Mineral Listing H-M Symbol (6/m) Hexagonal Ditrigonal Dipyramidal Mineral Listing H-M Symbol (6 m2) Hexagonal Hexagonal Crystallographic Pyramidal Mineral Listing Axes H-M Symbol (6) Hexagonal Trapezohedral Mineral Listing H-M Symbol (6 2 2) Hexagonal Trigonal Dipyramidal Mineral H-M Symbol (6) The Hexagonal System (Trigonal Division) Trigonal Minerals Crystal Form Example. Class Unknown Mineral Listing. , , , , Trigonal Ditrigonal Pyramidal Mineral Listing and Stereo image. H-M Symbol (3m) Trigonal Hexagonal Scalenohedral Mineral Listing and Stereo image. H-M Symbol (3 2/m) Trigonal Pyramidal Mineral Listing and Stereo image. Trigonal H-M Symbol (3) Crystallographic Axes Trigonal Rhombohedral Mineral Listing and Stereo image. H-M Symbol ( 3) Trigonal Trapezohedral Mineral Listing and Stereo image. H-M Symbol (3 2) The Monoclinic System Monoclinic Minerals Crystal Form Example. Class Unknown Mineral Listing. , , , , Monoclinic Domatic Mineral Listing H-M Symbol (m) Monoclinic Prismatic Mineral Listing H-M Symbol (2/m) Monoclinic Crystallographic Axes b 90º Monoclinic Sphenoidal Mineral Listing H-M Symbol (2) The Triclinic System Triclinic Minerals Crystal Form Class Unknown M Example. ineral , , , , Listing. Triclinic Pedial Mineral Listing H-M Symbol (1) Triclinic Crystallographic Triclinic Axes Pinacoidal Miner a,b,g 90 al Listing H-M Symbol ( 1) Open Forms and Closed Forms  A closed form is a set of crystal faces that completely enclose space. Thus, in crystal classes that contain closed forms, a crystal can be made up of a single form.  An open form is one or more crystal faces that do not completely enclose space.  Example 1. Pedions are single faced forms. Since there is only one face in the form a pedion cannot completely enclose space. Thus, a crystal that has only pedions, must have at least 3 different pedions to completely enclose space.  Example 2. A prism is a 3 or more faced form wherein the crystal faces are all parallel to the same line. If the faces are all parallel then they cannot completely enclose space. Thus crystals that have prisms must also have at least one additional form in order to completely enclose space.  Example 3. A dipyramid has at least 6 faces that meet in points at opposite ends of the crystal. These faces can completely enclose space, so a dipyramid is closed form. Although a crystal may be made up of a single dipyramid form, it may also have other forms present. Pedions  A pedion is an open, one faced form. Pedions are the only forms that occur in the Pedial class (1). Since a pedion is not related to any other face by symmetry, each form symbol refers to a single face. For example the form {100} refers only to the face (100), and is different from the form {00} which refers only to the face (00). Note that while forms in the Pedial class are pedions, pedions may occur in other crystal classes. Pinacoids  A Pinacoid is an open 2-faced form made up of two parallel faces. In the crystal drawing shown here the form {111} is a pinacoid and consists of two faces, (111) and (). The form {100} is also a pinacoid consisting of the two faces (100) and (00). Similarly the form {010} is a pinacoid consisting of the two faces (010) and (00), and the form {001} is a two faced form consisting of the faces (001) and (00). In this case, note that at least three of the above forms are necessary to completely enclose space. While all forms in the Pinacoid class are pinacoids, pinacoids may occur in other crystal classes as well. Domes  Domes are 2- faced open forms where the 2 faces are related to one another by a mirror plane. In the crystal model shown here, the dark shaded faces belong to a dome. The vertical faces along the side of the model are pinacoids (2 parallel faces). The faces on the front and back of the model are not related to each other by symmetry, and are thus two different pedions. Sphenoids  Sphenoids are2 - faced open forms where the faces are related to each other by a 2-fold rotation axis and are not parallel to each other. The dark shaded triangular faces on the model shown here belong to a sphenoid. Pairs of similar vertical faces that cut the edges of the drawing are also pinacoids. The top and bottom faces, however, are two different pedions. Prisms  A prism is an open form consisting of three or more parallel faces. Depending on the symmetry, several different kinds of prisms are possible Trigonal prism  3 - faced form with all faces parallel to a 3 -fold rotation axis Ditrigonal prism  6 - faced form with all 6 faces parallel to a 3-fold rotation axis. Note that the cross section of this form (shown to the right of the drawing) is not a hexagon, i.e. it does not have 6-fold rotational symmetry Rhombic prism  4 - faced form with all faces parallel to a line that is not a symmetry element. In the drawing to the right, the 4 shaded faces belong to a rhombic prism. The other faces in this model are pinacoids (the faces on the sides belong to a side pinacoid, and the faces on the top and bottom belong to a top/bottom pinacoid). Tetragonal prism  4 - faced open form with all faces parallel to a 4-fold rotation axis or. The 4 side faces in this model make up the tetragonal prism. The top and bottom faces make up the a form called the top/bottom pinacoid. Ditetragonal prism  8 - faced form with all faces parallel to a 4-fold rotation axis. In the drawing, the 8 vertical faces make up the ditetragonal prism. Hexagonal prism  6 - faced form with all faces parallel to a 6-fold rotation axis. The 6 vertical faces in the drawing make up the hexagonal prism. Again the faces on top and bottom are the top/bottom pinacoid form Dihexagonal prism  12 - faced form with all faces parallel to a 6-fold rotation axis. Note that a horizontal cross-section of this model would have apparent 12-fold rotation symmetry. The dihexagonal prism is the result of mirror planes parallel to the 6-fold rotation axis. Pyramids  A pyramid is a 3, 4, 6, 8 or 12 faced open form where all faces in the form meet, or could meet if extended, at a point. Trigonal pyramid  3-faced form where all faces are related by a 3-fold rotation axis. Ditrigonal pyramid  6-faced form where all faces are related by a 3-fold rotation axis. Note that if viewed from above, the ditrigonal pyramid would not have a hexagonal shape; its cross section would look more like that of the trigonal prism discussed above Rhombic pyramid  4-faced form where the faces are related by mirror planes. In the drawing shown here the faces labeled "p" are the four faces of the rhombic pyramid. If extend, these 4 faces would meet at a point. Tetragonal pyramid  4-faced form where the faces are related by a 4 axis. In the drawing the small triangular faces that cut the corners represent the tetragonal pyramid. Note that if extended, these 4 faces would meet at a point. Ditetragonal pyramid  8-faced form where all faces are related by a 4 axis. In the drawing shown here, the upper 8 faces belong to the ditetragonal pyramid form. Note that the vertical faces belong to the ditetragonal prism. Hexagonal pyramid  6-faced form where all faces are related by a 6 axis. If viewed from above, the hexagonal pyramid would have a hexagonal shape. Dihexagonal pyramid  12-faced form where all faces are related by a 6-fold axis. This form results from mirror planes that are parallel to the 6-fold axis. Dipyramids  Dipyramids are closed forms consisting of 6, 8, 12, 16, or 24 faces. Dipyramids are pyramids that are reflected across a mirror plane. Thus, they occur in crystal classes that have a mirror plane perpendicular to a rotation or rotoinversion axis. Trigonal dipyramid  6-faced form with faces related by a 3-fold axis with a perpendicular mirror plane. In this drawing, all six faces belong to the trigonal-dipyramid Ditrigonal -dipyramid  12-faced form with faces related by a 3-fold axis with a perpendicular mirror plane. If viewed from above, the crystal will not have a hexagonal shape, rather it would appear similar to the horizontal cross-section of the ditrigonal prism, discussed above. Ditrigonal -dipyramid  12-faced form with faces related by a 3-fold axis with a perpendicular mirror plane. If viewed from above, the crystal will not have a hexagonal shape, rather it would appear similar to the horizontal cross-section of the ditrigonal prism, discussed above Rhombic dipyramid  8-faced form with faces related by a combinations of 2-fold axes and mirror planes. The drawing to the right shows 2 rhombic dipyramids. One has the form symbol {111} and consists of the four larger faces shown plus four equivalent faces on the back of the model. The other one has the form symbol {113} and consists of the 4 smaller faces shown plus the four on the back. Tetragonal dipyramid  8-faced form with faces related by a 4-fold axis with a perpendicular mirror plane. The drawing shows the 8-faced tetragonal dipyramid. Also shown is the 4-faced tetragonal prism, and the 2-faced top/bottom pinacoid. Ditetragonal dipyramid  16-faced form with faces related by a 4-fold axis with a perpendicular mirror plane. The ditetragonal dipyramid is shown here. Note the vertical faces belong to a ditetragonal prism. Hexagonal dipyramid  12-faced form with faces related by a 6-fold axis with a perpendicular mirror plane. The vertical faces in this model make up a hexagonal prism. Dihexagonal dipyramid  24-faced form with faces related by a 6-fold axis with a perpendicular mirror plane. Trapezohedrons  Trapezohedron are closed 6, 8, or 12 faced forms, with 3, 4, or 6 upper faces offset from 3, 4, or 6 lower faces. The trapezohedron results from 3-, 4-, or 6-fold axes combined with a perpendicular 2-fold axis. An example of a tetragonal trapezohedron is shown in the drawing to the right. Other examples are shown in your textbook. Scalenohedrons  A scalenohedron is a closed form with 8 or 12 faces. In ideally developed faces each of the faces is a scalene triangle. In the model, note the presence of the 3-fold rotoinversion axis perpendicular to the 3 2-fold axes. Rhombohedrons  A rhombohedron is 6-faced closed form wherein 3 faces on top are offset by 3 identical upside down faces on the bottom, as a result of a 3-fold rotoinversion axis. Rhombohedrons can also result from a 3-fold axis with perpendicular 2-fold axes. Rhombohedrons only occur in the crystal classes 2/m , 32, and. Disphenoids  A disphenoid is a closed form consisting of 4 faces. These are only present in the orthorhombic system (class 222) and the tetragonal system (class ) Hexahedron  A hexahedron is the same as a cube. 3-fold axes are perpendicular to the face of the cube, and four axes run through the corners of the cube. Note that the form symbol for a hexahedron is {100}, and it consists of the following 6 faces: (100), (010), (001), (00), (00), and (00). Octahedron  An octahedron is an 8 faced form that results form three 4-fold axes with perpendicular mirror planes. The octahedron has the form symbol {111}and consists of the following 8 faces: (111), (), (11), (1), (1), (1), (11), and (11). Note that four 3-fold axes are present that are perpendicular to the triangular faces of the octahedron (these 3-fold axes are not shown in the drawing). Dodecahedron  A dodecahedron is a closed 12-faced form. Dodecahedrons can be formed by cutting off the edges of a cube. The form symbol for a dodecahedron is {110}. As an exercise, you figure out the Miller Indices for these 12 faces. Tetrahexahedron  The tetrahexahedron is a 24-faced form with a general form symbol of {0hl} This means that all faces are parallel to one of the a axes, and intersect the other 2 axes at different lengths. Trapezohedron  An isometric trapezohedron is a 12-faced closed form with the general form symbol {hhl}. This means that all faces intersect two of the a axes at equal length and intersect the third a axis at a different length. Tetrahedron  The tetrahedron occurs in the class 3m and has the form symbol {111}(the form shown in the drawing) or {11} (2 different forms are possible). It is a four faced form that results form three axes and four 3-fold axes (not shown in the drawing). Gyroid  A gyroid is a form in the class 432 (note no mirror planes) Pyritohedron  The pyritohedron is a 12-faced form that occurs in the crystal class 2/m. Note that there are no 4-fold axes in this class. The possible forms are {h0l} or {0kl} and each of the faces that make up the form have 5 sides. Diploid  The diploid is the general form {hkl} for the diploidal class (2/m). Again there are no 4-fold axes. Tetartoid  Tetartoids are general forms in the tetartoidal class (23) which only has 3-fold axes and 2-fold axes with no mirror planes.  Introduction to Crystallography and Mineral Crystal Systems Crystal Forms and Symmetry Classes CRYSTAL FORMS and the 32 SYMMETRY CLASSES! The term "form" is loosely used by many people to indicate outward appearance. However, we must "tighten up" our definition when discussing crystallography. HABIT is the correct term to indicate outward appearance. Habit, when applied to natural crystals and minerals, includes such descriptive terms as tabular, equidimensional, acicular, massive, reniform, drusy, and encrusting. Drusy Quartz in Geode Tabular Orthoclase Feldspar Encrusting Smithsonite A FORM is a group of crystal faces, all having the same relationship to the elements of symmetry of a given crystal system. These crystal faces display the same physical and chemical properties because the ATOMIC ARRANGEMENT (internal geometrical relationships) of the atoms composing them is the SAME. The relationship between form and the elements of symmetry is an important one to grasp, because no matter how distorted a natural crystal may be, certain key elements will be recognizable to help the student discern what form or forms are present. The term general form has specific meaning in crystallography. In each crystal class, there is a form in which the faces intersect each crytallographic axes at different lengths. This is the general form {hkl} and is the name for each of the 32 classes (hexoctahedral class of the isometric system, for example). All other forms are called special forms. Let's look at an octahedron as an example (fig. 2.1). All the crystal faces present are the expression of the repetition of a single form having the Miller indices of {111} about the three crystallographic axes (remember those from the first article?). Each face on a natural crystal (octahedral galena or fluorite are examples), when rotated to the position of the (111) face in the drawing, would have the same shape and orientation of striations, growth pits or stair steps, and etch pits, if present. The presence of these features is true whether or not the crystal is well formed or distorted in its growth. Note that I did NOT state that the faces are necessarily the same size on the natural crystal! In fact, due to variations in growth conditions, the faces are usually not. In the literature, you may see a notation, given as {hkl}. This is the notation, presented as Miller indices for general form. The octahedral form 1 is given as {111}, the same as the face that intersects all positive ends of the crystallographic axes. A single form may show closure, as with an octahedron, or may not, as in a pinacoid (an open two-faced form). So every form has an {hkl} notation. In the case of general notation concerning the hexagonal system, it is {hk-il} and is read as "h, k, minus i, l". Before leaving this discussion of form, here are a couple of examples of how knowledge of the interrelationships of forms and crystal systems may be used. Someone gives you a quartz crystal and says, "Look at this crystal. It's not normal." Normal to this person, we assume means an elongate (prismatic), 6-sided crystal form with a 6-faced termination on the free-growth end. When you examine the "abnormal" crystal, it is highly distorted, broken, and has regrowth faces. Prism faces on quartz crystals almost always have striations at right angles to the c crystallographic axis and parallel to the plane of the a1, a2, and a3 axes. These striations are due to the variable growth rates of the terminal faces as the mineral crystallized. Knowing about the striations and their orientation, you examine the surface of the specimen and, with reflected light, find the prism faces by their striations. Terminal (or pyramidal) faces on quartz crystals often exhibit triangular pits or platforms. By finding them, you can then determine if any other faces that would be really unusual are present. Not finding any unusual faces, you can return the specimen to the person with the comment, "Well, your crystal is certainly interesting, but it does not have any unusual forms. What it does display is a complex growth history reflected by its less than ideal crystal shape." Most people and many collectors recognize unusual habits, but not unusual forms. They note that the shape of a crystal is odd looking, but don't have the background in crystallography to know if the crystal is truly unusual. A broken and regrown quartz crystal is not particularly special, but a quartz crystal with a c pinacoidal termination is worth noting, as it is a very uncommon form for quartz. I have only seen a few from one locality. Being the skeptic that I am, I purchased one crystal which had another mineral coating the termination. I mechanically removed the coating mineral with the edge of a pocket knife. There for my examination was the c pinacoid termination {0001}, satisfying me that it was a natural growth form! A crystal's form may be completely described by use of the Miller's indices and the Hermann-Mauguin notation of its POINT GROUP SYMMETRY. The latter notation tells us how to orient the crystal, in each specific crystal class, to recognize which axis (a, b, or c) is designated as having the highest symmetry. It also tells us what other symmetry elements may be present and where they are in orientation to the other elements. 2 Types of Crystal Forms Note that there are TWO GENERAL TYPES OF FORMS: those that by repetition close on themselves creating a complete form (termed closed) and those that do not (termed open). There are 32 (some say 33) forms in the non-isometric (noncubic) crystal systems and another 15 forms in the isometric (cubic) system. Let's start to familiarize ourselves with them by making a tabulation and including the number of faces (below). Isometric Crystal Forms Numbe Number r Name Name of of Faces Faces (1) Cube 6 9)Tristetrahedron 12 (2) Octahedron 8 (10) Hextetrahedron 24 (11) Deltoid (3) Dodecahedron 12 dodecahedron 24 (4) (12) Gyroid Tetrahexahedron 24 24 (5) Trapezohedron 24 (13) Pyritohedron 12 (6) Trisoctahedron 24 (14) Diploid 24 (7) Hexoctahedron 48 (15) Tetartoid 12 (8) Tetrahedron 4 3  Non-Isometric Crystal Forms Numbe Number r Name Name of of Faces Faces (16) Pedion* 1 (32) Dihexagonal pyramid 12 (17) Pinacoid** 2 (33) Rhombic dipyramid 8 (18) Dome or Sphenoid 2 (34) Trigonal dipyramid 6 (19) Rhombic prism 4 (35) Ditrigonal dipyramid 12 (20) Trigonal prism 3 (36) Tetragonal dipyramid 8 (21) Ditrigonal prism 6 (37) Ditetragonal dipyramid 16 (22) Tetragonal prism 4 (38) Hexagonal dipyramid 12 (23) Ditetragonal prism 8 (39) Dihexagonal dipyramid 24 (40) Trigonal (24) Hexagonal prism 6 trapezohedron 6 (41) Tetragonal (25) Dihexagonal prism 12 trapezohedron 8 (42) Hexagonal (26) Rhombic pyramid 4 trapezohedron 12 (43)Tetragonal (27) Trigonal pyramid 3 scalenohedron 8 4  (44) Hexagonal (28)Ditrigonal pyramid 6 scalenohedron 12 (29) Tetragonal (45) Rhombohedron pyramid 4 6 (30) Ditetragonal (46) Rhombic disphenoid pyramid 8 4 (31) Hexagonal (47) Tetragonal disphenoid pyramid 6 4 *Pedion may appear in several crystal systems **Pinacoid drawing displays 3 pairs of pinacoid faces from the Orthorhombic system. Pinacoids appear in several crystal systems. 5 Crystal form, in crystallography, all crystal faces having similar symmetry. Those forms that enclose space are called closed forms; those that do not, open forms. The faces that comprise a form will be similar in appearance, even though of different shapes and sizes; this similarity may be evident from natural striations, etchings, or growths, or it may be apparent only after etching with acid. The forms in all crystal systems except the isometric are similar and may be generally described as follows: 1. Pedion: a single face; 2. Pinacoid: pair of opposite faces parallel to two of the principal crystallographic axes; 3. Dome: two nonparallel faces symmetrical to a plane of symmetry; 4. Sphenoid: two nonparallel faces symmetrical to a 2- or 4-fold axis of symmetry; 5. Disphenoid: four-faced closed form in which the two faces of a sphenoid alternate above two faces of another sphenoid; 6. Prism: 3, 4, 6, 8, or 12 faces the intersection lines of which are parallel and (except for some monoclinic prisms) are parallel to a principal crystallographic axis; 7. Pyramid: 3, 4, 6, 8, or 12 nonparallel faces that meet in a point; 8. Scalenohedron: 8-faced (tetragonal) or 12-faced (hexagonal) closed form in which the faces are grouped in symmetrical pairs; in perfect crystals, each face is a scalene triangle; 9. Trapezohedron: 6-, 8-, 12-, or 24-faced closed form in which half the faces are offset above the other half; in well-developed crystals, each face is a trapezium; 10. Dipyramid: 6-, 8-, 12-, 16-, or 24-faced closed form in which the lower pyramid is a reflection of the upper; 1 11. Rhombohedron: closed form of six identical faces in which none of the intersection edges is perpendicular. These open forms can NOT form a whole crystal by themselves and need other forms to finish the crystal. Closed forms can form a whole crystal completely by themselves. 2 Crystal Systems Every crystal class is a member of one of the six crystal systems. These systems include the isometric, hexagonal, tetragonal, orthorhombic, monoclinic, and triclinic crystal systems. The hexagonal crystal system is further broken down into hexagonal and rhombohedral divisions. Every crystal class which belongs to a certain crystal system will share a characteristic symmetry element with the other members of its system. For example, all crystals of the isometric system possess four 3-fold axes of symmetry which proceed diagonally from corner to corner through the center of the cubic unit cell. In contrast, all crystals of the hexagonal division of the hexagonal system possess a single six-fold axis of rotation. In addition to the characteristic symmetry element, a crystal class may possess other symmetry elements which are not necessarily present in all members of the same system. The crystal class which possesses the highest possible symmetry or the highest number of symmetry elements within each system is termed the holomorphic class of the system. For example, crystals of the holomorphic class of the isometric system possess inversion symmetry, three 4-fold axes of rotational symmetry, the characteristic set of four 3-fold axes of rotational symmetry which is indicative of the isometric crystal system, six 2-fold axes of rotational symmetry, and nine different mirror planes. In contrast, a crystal which is not a member of the holomorphic class yet still belongs to the isometric system may possess only three 2-fold axes of rotational symmetry and the characteristic four 3-fold axes of rotational symmetry. The crystal system of a mineral species may sometimes be determined in the field by visually examining a particularly well-formed crystal of the species. 1 Isometric The isometric crystal system is also known as the cubic system. The crystallographic axes used in this system are of equal length and are mutually perpendicular, occurring at right angles to one another. All crystals of the isometric system possess four 3-fold axes of symmetry, each of which proceeds diagonally from corner to corner through the center of the cubic unit cell. Crystals of the isometric system may also demonstrate up to three separate 4-fold axes of rotational symmetry. These axes, if present, proceed from the center of each face through the origin to the center of the opposite face and correspond to the crystallographic axes. Furthermore crystals of the isometric system may possess six 2-fold axes of symmetry which extend from the center of each edge of the crystal through the origin to the center of the opposite edge. Minerals of this system may demonstrate up to nine different mirror planes. Examples of minerals which crystallize in the isometric system are halite, magnetite, and garnet. Minerals of this system tend to produce crystals of equidimensional or equant habit. Hexagonal Minerals of the hexagonal crystal system are referred to three crystallographic axes which intersect at 120° and a fourth which is perpendicular to the other three. This fourth axis is usually depicted vertically. The hexagonal crystal system is divided into the hexagonal and rhombohedral or trigonal divisions. All crystals of the hexagonal division possess a single 6-fold axis of rotation. In addition to the single 6-fold axis of rotation, crystals of the hexagonal division may possess 2  up to six 2-fold axes of rotation. They may demonstrate a center of inversion symmetry and up to seven mirror planes. Crystals of the trigonal division all possess a single 3-fold axis of rotation rather than the 6-fold axis of the hexagonal division. Crystals of this division may possess up to three 2-fold axes of rotation and may demonstrate a center of inversion and up to three mirror planes. Minerals species which crystallize in the hexagonal division are apatite, beryl, and high quartz. Minerals of this division tend to produce hexagonal prisms and pyramids. Example species which crystallize in the rhombohedral division are calcite, dolomite, low quartz, and tourmaline. Such minerals tend to produce rhombohedra and triangular prisms. Tetragonal Minerals of the tetragonal crystal system are referred to three mutually perpendicular axes. The two horizontal axes are of equal length, while the vertical axis is of different length and may be either shorter or longer than the other two. Minerals of this system all possess a single 4-fold symmetry axis. They may possess up to four 2- fold axes of rotation, a center of inversion, and up to five mirror planes. Mineral species which crystallize in the tetragonal crystal system are zircon and cassiterite. These minerals tend to produce short crystals of prismatic habit. Orthorhombic Minerals of the orthorhombic crystal system are referred to three mutually perpendicular axes, each of which is of a different length than the others. 3  Crystals of this system uniformly possess three 2-fold rotation axes and/or three mirror planes. The holomorphic class demonstrates three 2-fold symmetry axes and three mirror planes as well as a center of inversion. Other classes may demonstrate three 2-fold axes of rotation or one 2-fold rotation axis and two mirror planes. Species which belong to the orthorhombic system are olivine and barite. Crystals of this system tend to be of prismatic, tabular, or acicular habit. Monoclinic Crystals of the monoclinic system are referred to three unequal axes. Two of these axes are inclined toward each other at an oblique angle; these are usually depicted vertically. The third axis is perpendicular to the other two. The two vertical axes therefore do not intersect one another at right angles, although both are perpendicular to the horizontal axis. Monoclinic crystals demonstrate a single 2-fold rotation axis and/or a single mirror plane. The holomorphic class possesses the single 2- fold rotation axis, a mirror plane, and a center of symmetry. Other classes display just the 2-fold rotation axis or just the mirror plane. Mineral species which adhere to the monoclinic crystal system include pyroxene, amphibole, orthoclase, azurite, and malachite, among many others. The minerals of the monoclinic system tend to produce long prisms. Triclinic Crystals of the triclinic system are referred to three unequal axes, all of which intersect at oblique angles. None of the axes are perpendicular to any other axis. Crystals of the triclinic system may be said to possess only a 1-fold 4  symmetry axis, which is equivalent to possessing no symmetry at all. Crystals of this system possess no mirror planes. The holomorphic class demonstrates a center of inversion symmetry. Mineral species of the triclinic class include plagioclase and axinite; these species tend to be of tabular habit. Crystal Forms A crystal form is a set of faces which are geometrically equivalent and whose spatial positions are related to one another according to the symmetry of the crystal. If one face of a crystal form is defined, the point symmetry operations which specify the class to which the crystal belongs also determine the other faces of the crystal form. Fifteen different forms are possible within the isometric or cubic system. These include the hexoctahedron, gyroid, hextetrahedron, diploid, and tetartoid, among others. The crystal forms of the remaining five crystal systems are the monohedron or pedion, parallelohedron or pinacoid, dihedron, or dome and sphenoid, disphenoid, prism, pyramid, dipyramid, trapezohedron, scalenohedron, rhombohedron, and tetrahedron. The crystal forms which occur in each crystal class and system must possess a symmetry complementary to that of the associated crystal class and system. For example, a monohedron, which possesses only one face, will never occur in a crystal with inversion symmetry because the inversion operation requires that an equivalent face be present on the opposite side of the crystal. A simple crystal may consist of only a single crystal form. A more complicated crystal may be a combination of several different forms. All 5 forms which occur in a crystal of a particular system must be compatible with that crystal system. Monohedron The monohedral crystal form is also called a pedion. It consists of a single face which is geometrically unique for the crystal and is not repeated by any set of symmetry operations. Members of the triclinic crystal system produce monohedral crystal forms. Parallelohedron The parallelohedral crystal form is also called a pinacoid. It consists of two and only two geometrically equivalent faces which occupy opposite sides of a crystal. The two faces are parallel and are related to one another only by a reflection or an inversion. Members of the triclinic crystal system produce parallelohedral crystal forms. Dihedron The dihedron consists of two and only two nonparallel geometrically equivalent faces. The two faces may be related by a reflection or by a rotation. The dihedron is termed a dome if the two faces are related only by reflection across a mirror plane. If the two faces are related instead by a 2-fold rotation axis then the dihedron is termed a sphenoid. Members of the monoclinic crystal system produce dihedral crystal forms. Disphenoid Members of the orthorhombic and tetragonal crystal systems produce rhombic and tetragonal disphenoids, which possess two sets of nonparallel geometrically equivalent faces, each of which is related by a 2-fold rotation. The faces of the upper sphenoid alternate with the faces of the lower sphenoid in such forms. Prism A prism is composed of a set of 3, 4, 6, 8, or 12 geometrically equivalent faces which are all parallel to the same axis. Each of these 6  faces intersects with the two faces adjacent to it to produce a set of parallel edges. The mutually parallel edges of all intersections of the prism sides then form a tube. Prisms are given names based on the shape of their cross section. Variants of the prism form include the rhombic prism, tetragonal prism, trigonal prism, and hexagonal prism. A prism in which the large faces are divided into two mirror-image faces which intersect with one another at an oblique angle is called a ditetragonal prism, a ditrigonal prism, or a dihexagonal prism. Prisms are associated with the members of the monoclinic crystal system. Pyramid A pyramid is composed of a set of 3, 4, 6, 8, or 12 faces which are not parallel but instead intersect at a point. The orthorhombic, tetragonal and hexagonal crystal systems all produce pyramids. These pyramids are named according to the shape of their cross-section in the same way that prisms are. Thus are produced the rhombic pyramid, tetragonal pyramid, trigonal pyramid, and hexagonal pyramid. Each large face of the ditetragonal pyramid, ditrigonal pyramid, and dihexagonal pyramids is divided into two mirror-image faces which occupy an oblique angle with respect to one another. Dipyramid The dipyramidal crystal form is composed of two pyramids placed base-to-base and related by reflection across a mirror plane which runs parallel to and adjacent to the pyramid bases. The upper and lower pyramids may each have 3, 4, 6, 8, or 12 faces; the dipyramidal form therefore possesses a total of 6, 8, 12, 16, or 24 faces. The orthorhombic, tetragonal and hexagonal crystal systems all produce dipyramids. These dipyramids are named for the shape of their cross- 7  section just as prisms and pyramids are, resulting in the rhombic dipyramid, trigonal dipyramid, tetragonal dipyramid, and hexagonal dipyramid. The large faces of the ditetragonal, ditrigonal and dihexagonal dipyramids are divided into two mirror-image faces which intersect one another at an oblique angle. Trapezohedron A trapezohedron is a crystal form possessing 6, 8, or 12 trapezoidal faces. The tetragonal crystal system and both the trigonal and hexagonal divisions of the hexagonal crystal system produce trapezohedral crystal forms. Trigonal trapezohedra possess three trapezoidal faces on the top and three on the bottom for a total of six faces; tetragonal trapezohedra have four faces on top and four on the bottom for a total of eight faces; and hexagonal trapezohedra have six faces on top and six on the bottom, resulting in twelve faces total. Scalenohedron A scalenohedron consists of 8 or 12 faces, each of which is a scalene triangle. The faces appear to be grouped into symmetric pairs. The tetragonal and hexagonal crystal systems produce the scalenohedral crystal form, of which examples may be further described as trigonal, tetragonal and hexagonal scalenohedra. Rhombohedron The rhombohedral crystal form possesses six rhombus-shaped faces. A rhombohedron resembles in appearance a cube which is poised upright upon one corner and has been either flattened or elongated along an axis which runs diagonally from corner to corner through the center. The rhombohedral crystal form is produced only by members of the trigonal and rhombohedral divisions of the hexagonal crystal system. 8 Tetrahedron A tetrahedron is composed of four triangular faces. In crystals of the isometric system each face is an identical equilateral triangle. In crystals of the tetragonal system each face is an identical isoceles triangle; this variant of the tetrahedron is called a tetragonal tetrahedron. In crystals of the orthorhombic system the faces consist of two pairs of different isoceles triangles; the crystal is then termed a rhombic tetrahedron. Crystal Classes The reflection, rotation, inversion, and rotoinversion symmetry operations may be combined in thirty-two different ways. Thirty-two different crystal classes are therefore defined so that each crystal class corresponds to a unique set of symmetry operations. Each of the crystal classes is named according to the variant of a crystal form which it displays. For example, the isometric hexoctahedral class belongs to the isometric crystal system and demonstrates the hexoctahedral crystal form. The rhombic pyramidal, tetragonal pyramidal, trigonal pyramidal and hexagonal pyramidal classes each display a variant of the crystal form which is called a pyramid. Each crystal class is a member of one of the six different crystal systems according to which characteristic symmetry operation it possesses. For example, all crystals of the isometric system possess four 3-fold axes of 9 symmetry, while minerals of the tetragonal system possess a single 4-fold symmetry axis and crystals of the triclinic class show no symmetry at all. The rhombic pyramidal crystal class is thus a member of the orthorhombic crystal system, the tetragonal pyramidal class is a member of the tetragonal crystal system, and the trigonal and hexagonal pyramidal classes are members of the rhombohedral (trigonal) and hexagonal divisions of the hexagonal crystal system respectively. Table of the 32 Crystal Classes The following table lists in bold type the six crystal systems. Included are the isometric, hexagonal, tetragonal, orthorhombic, monoclinic, and triclinic systems. The tetragonal crystal system is further separated into the hexagonal and trigonal or rhombohedral divisions. Under each crystal system the table lists by name the crystal classes which occur within that system. For example, the crystal classes which occur within the trigonal crystal system are the trigonal monohedral and trigonal parallelohedral crystal classes. Adjacent to the listing of each crystal class is the symmetry of the class. When listing the symmetry of each crystal class an axis of rotational symmetry is represented by the capital letter A. Whether this axis is a 2-fold, 3-fold, or 4-fold axis is indicated by a subscript following the letter A. The number of such axes present is indicated by a numeral preceding the capital A. 1A2, 2A3, and 3A4 thus represent one 2-fold axis of rotation, two 3-fold axes, and three 4-fold axes respectively. A center of inversion is noted by the lowercase letter 'i' while a mirror plane is denoted by 'm'. The numeral preceding the m indicates how many mirror planes are present. Axes of rotary inversion are usually replaced by the equivalent rotations and reflections. For example, a 2-fold rotoinversion axis is equivalent to reflection through a mirror plane perpendicular to the rotoinversion axis. A crystal 10 which possesses a 3-fold rotoinversion axis is equivalent to one which possesses both 3-fold rotational symmetry and inversion symmetry. A 6-fold rotoinversion is equivalent to 3-fold rotation and reflection across a mirror plane at right angles to the rotation axis. The only rotoinversion operation which cannot be thus replaced is 4-fold rotoinversion, which is indicated by R4. The class which possesses the highest possible symmetry within each crystal system is termed the holomorphic class of that system. The holomorphic class of each crystal system is indicated in the table by bold type. For example, the triclinic parallelohedron is the holomorphic class of the triclinic crystal system while the isometric hexoctahedron is the holomorphic class of the isomorphic or cubic crystal system. The characteristic symmetry element of each crystal system is listed in bold type. It is thus apparent that the characteristic symmetry element of the isometric crystal system is the possession of four 3-fold axes of rotational symmetry, while the characteristic symmetry element of the rhombohedral division of the hexagonal crystal system is the possession of a single 3-fold axis of rotational symmetry. Crystal Class / Symmetry of Crystal System Crystal Form Class i, 3A4, 4A3, 6A2, hexoctahedron 9m gyroid Isometric 3A4, 4A3, 6A2 hextetrahedron System 3A2, 4A3, 6m diploid i, 3A2, 4A3, 3m tetartoid 3A2, 4A3 11  dihexagonal dipyramid i, 1A6, 6A2, 7m hexagonal 1A6, 6A2 trapezohedron 1A6, 6m Hexagonal Hexagonal dihexagonal pyramid 1R6, 3A2, 3m System Division ditrigonal dipyramid i, 1A6, 1m hexagonal dipyramid 1A6 hexagonal pyramid 1R6 trigonal dipyramid hexagonal scalenohedron i, 1A3, 3A2, 3m trigonal trapezohedron 1A3, 3A2 Rhombohedral 1A3, 3m Division ditrigonal pyramid i, 1A3 rhombohedron 1A3 trigonal pyramid ditetragonal dipyramid i, 1A4, 4A2, 5m tetragonal 1A4, 4A2 trapzohedron 1A4, 4m Tetragonal ditetragonal pyramid 1R4, 2A2, 2m System tetragonal i, 1A4, 1m scalenohedron 1A4 tetragonal dipyramid 1R4 tetragonal pyramid tetragonal disphenoid Orthorhombic rhombic dipyramid i, 3A2, 3m System rhombic disphenoid 3A2 12  rhombic pyramid 1A2, 2m prism i, 1A2, 1m Monoclinic sphenoid 1A2 System dome 1m parallellohedron i Triclinic System monohedron no symmetry 13 The 48 Special Crystal Forms. Forms, Open and Closed Any group of crystal faces related by the same symmetry is called a form. There are 47 or 48 crystal forms depending on the classification used. Closed forms are those groups of faces all related by symmetry that completely enclose a volume of space. It is possible for a crystal to have faces entirely of one closed form. Open forms are those groups of faces all related by symmetry that do not completely enclose a volume of space. A crystal with open form faces requires additional faces as well. There are 17 or 18 open forms and 30 closed forms. Triclinic, Monoclinic and Orthorhombic Forms Pedion A single face unrelated to any other by symmetry. Open Pinacoid A pair of parallel faces related by mirror plane or twofold symmetry axis. Open Dihedron A pair of intersecting faces related by mirror plane or twofold symmetry axis. Some crystallographers distinguish between domes (pairs of intersecting faces related by mirror plane) and sphenoids (pairs of intersecting faces related by twofold symmetry axis). All are open forms Pyramid A set of faces related by symmetry and meeting at a common point. Open form. 3-, 4- and 6-Fold Prisms Prism A collection of faces all parallel to a symmetry axis. All are open. 3-, 4- and 6-Fold Pyramids Pyramid A group of faces intersecting at a symmetry axis. All are open. The base of the pyramid would be a pedion. 3-, 4- and 6-Fold Dipyramids Dipyramid Two pyramids joined base to base along a mirror plane. All are closed, as are all following forms. Scalenohedra and Trapezohedra Disphenoid A solid with four congruent triangle faces, like a distorted tetrahedron. Midpoints of edges are twofold symmetry axes. In the tetragonal disphenoid the faces are isoceles triangles and a fourfold inversion axis joins the midpoints of the bases of the isoceles triangles. Scalenohedron A solid made up of scalene triangle faces (all sides unequal) Trapezohedron A solid made of trapezia (irregular quadrilaterals) Rhombohedron A solid with six congruent parallelogram faces. Can be considered a cube distorted along one of its diagonal three-fold symmetry axes. Tetartoidal, Gyroidal and Diploidal Forms Tetartoid The general form for symmetry class 233. 12 congruent irregular pentagonal faces. The name comes from a Greek root for one-fourth because only a quarter of the 48 faces for full isometric symmetry are present. Gyroid The general form for symmetry class 432. 24 congruent irregular pentagonal faces. Diploid The general form for symmetry class 2/m3*. 24 congruent irregular quadrilateral faces. The name comes from a Latin root for half, because half of the 48 faces for full isometric symmetry are present. Pyritohedron Special form (hk0) of symmetry class 2/m3*. Faces are each perpendicular to a mirror plane, reducing the number of faces to 12 pentagonal faces. Although this superficially looks like the Platonic solid with 12 regular pentagon faces, these faces are not regular. Hextetrahedral Forms Tetrahedron Four equilateral triangle faces (111) Trapezohedral Tristetrahedron 12 kite-shaped faces (hll) Trigonal Tristetrahedron 12 isoceles triangle faces (hhl). Like an tetrahedron with a low triangular pyramid built on each face. Hextetrahedron 24 triangular faces (hkl) The general form. Hexoctahedral Forms Cube Six square faces (100). Octahedron Eight equilateral triangle faces (111) Rhombic Dodecahedron 12 rhombic faces (110) Trapezohedral Trisoctahedron 24 kite-shaped faces (hhl). Note that the Miller indices for the two trisoctahedra are the opposite of those for the tristetrahedra. Trigonal Trisoctahedron 24 isoceles triangle faces (hll). Like an octahedron with a low triangular pyramid built on each face. Tetrahexahedron 24 isoceles triangle faces (h0l). Like an cube with a low pyramid built on each face. Hexoctahedron 48 triangular faces (hkl) The general form Law of crystal symmetry | Solid State | Physical Chemistry https://www.youtube.com/watch?v=7GkuqcoGCU4 Unit 3.3 - Point Symmetry and Rotoinversions by Frank Hoffman https://www.youtube.com/watch?v=9MMKjO5HB-I Unit 1.8 - The Seven Crystal Systems by Frank Hoffman https://www.youtube.com/watch?v=_4OJKPoZn5o&list=PL6C90- 24AMSNIUNJTsOhNKMfPpvlC9FHu&index=9 Mineralogy: Lecture 6, Hermann-Mauguin Notation by Kenneth Befus https://www.youtube.com/watch?v=MnFFqAPca40