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Fig It is very easy to visualize the location of a simple face given miller index, or to derive a miller index from simple faces a b c -b -c

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Hexagonal Miller index There need to be 4 intercepts (hkil) There need to be 4 intercepts (hkil) h = a 1 h = a 1 k = a 2 k = a 2 i = a 3 i = a 3 l = c l = c Two a axes have to have opposite sign of other axis so that Two a axes have to have opposite sign of other axis so that h + k + i = 0 h + k + i = 0 Possible to report the index two ways: Possible to report the index two ways: (hkil) (hkil) (hkl) (hkl)

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Klein and Hurlbut Fig (100) (1010) (110)(111) (1120)(1121)

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Assigning Miller indices Prominent (and common) faces have small integers for Miller Indices Prominent (and common) faces have small integers for Miller Indices Faces that cut only one axis Faces that cut only one axis (100), (010), (001) etc (100), (010), (001) etc Faces that cut two axes Faces that cut two axes (110), (101), (011) etc (110), (101), (011) etc Faces that cut three axes Faces that cut three axes (111) (111) Called unit face Called unit face

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Zones, Forms, Habits Quantitative description of orientation in minerals – use Miller indices: Quantitative description of orientation in minerals – use Miller indices: Zone - Lines, or linear directions within minerals Zone - Lines, or linear directions within minerals Form - Shapes of three dimensional objects Form - Shapes of three dimensional objects Qualitative description of mineral shapes: Qualitative description of mineral shapes: Habit Habit

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Crystal Habit Qualitative terminology to describe individual minerals and aggregates of minerals Qualitative terminology to describe individual minerals and aggregates of minerals Shape of individual minerals Shape of individual minerals Intergrowths of several mineral grains Intergrowths of several mineral grains Shape of masses of grains Shape of masses of grains

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Colloform Colloform finely crystalline, concentric mineral layer finely crystalline, concentric mineral layer Globular – (spherulitic) Globular – (spherulitic) radiating, concentrically arranged acicular minerals radiating, concentrically arranged acicular minerals Reniform Reniform kidney shaped kidney shaped Botryoidal Botryoidal like a bunch of grapes like a bunch of grapes Mammillary Mammillary similar, but larger than botryoidal, breast-like or portions of spheres similar, but larger than botryoidal, breast-like or portions of spheres Drusy Drusy Surface covered with layer of small crystals Surface covered with layer of small crystals

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Globular hematite Drusy quartz

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Fig Terminology useful for describing general shapes of minerals (table-like) (asbestos: amphiboles and pyroxenes) (knife like – kyanite) (Mica)

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Bladed kyanite Al 2 SiO 5 Fibrous tremolite Fibrous tremolite: amphibole Ca(Mg,Fe) 5 Si 8 O 22 (OH) 2

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Zones Collection of common faces Collection of common faces Parallel to some common line Parallel to some common line Line called the zone axis Line called the zone axis Identified by index [hkl] Identified by index [hkl] Zone axis parallels intersection of edges of faces Zone axis parallels intersection of edges of faces

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Fig 2-30 a c b Faces = (110), (110), (110), (110) Zone axis intesects (001) lattice nodes = [001] Note typo in first edition Intersection of faces = [001] Zone -a -c -b

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Fig 2-27 Other linear crystallographic directions Other linear crystallographic directions Includes crystallographic axes Includes crystallographic axes Referenced to intersection of lattice nodes Referenced to intersection of lattice nodes For example: location of rotation axes or other linear features For example: location of rotation axes or other linear features Lattice node

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Form Formal crystallographic nomenclature of the shape of minerals Formal crystallographic nomenclature of the shape of minerals Description Description Collection of crystal faces Collection of crystal faces Related to each other by symmetry Related to each other by symmetry Identified by index: {hkl} Identified by index: {hkl} Values for h, k and l are determined by one of the faces Values for h, k and l are determined by one of the faces

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Example There are six faces in a cube (a kind of form): There are six faces in a cube (a kind of form): (100), (010), (001), (100), (010), (001) (100), (010), (001), (100), (010), (001) All faces parallel two axes and are perpendicular to one axis All faces parallel two axes and are perpendicular to one axis Form is written with brackets Form is written with brackets Uses miller index of one face Uses miller index of one face Generally positive face Generally positive face E.g., {001} E.g., {001} a b c (001) Isometric form {001} (010) (100)

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Possible to determine the shape of a form with: Possible to determine the shape of a form with: 1)Miller index of one face in form 2)Point symmetry of the crystal class The form is created by operating point symmetry on the initial face The form is created by operating point symmetry on the initial face Number of faces in a form depends on crystal class Number of faces in a form depends on crystal class

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Fig {011} form in crystal class with point symmetry 2/m 2/m 2/m (Orthorhombic) – called a rhombic prism – called a rhombic prism Rhombus – an equilateral parallelogram Prism – a crystal form whose faces are parallel to one axis Mirror parallel to (010)Mirror parallel to (001) Face parallel to a axis

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Triclinic system: Triclinic system: Point group (i.e. crystal class) = 1 Point group (i.e. crystal class) = 1 Symmetry content = (1A 1 ) Symmetry content = (1A 1 ) {111} has only 2 faces {111} has only 2 faces

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Isometric system: Isometric system: Point group (crystal class) = 4/m 3 2/m Point group (crystal class) = 4/m 3 2/m Symmetry content = 3A 4, 4A 3, 6A 2, 9m Symmetry content = 3A 4, 4A 3, 6A 2, 9m {111} has 8 faces {111} has 8 faces Form is an octahedron Form is an octahedron

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Isometric system Isometric system Point group (crystal class) = 4 Point group (crystal class) = 4 Symmetry content = 1A 4 Symmetry content = 1A 4 {111} has 4 faces {111} has 4 faces Form is a tetrahedron Form is a tetrahedron C b a (111)

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Two types of forms: Two types of forms: Open form – does not enclose a volume Open form – does not enclose a volume Closed form – encloses a volume Closed form – encloses a volume Minerals must have more than one form if they have an open form Minerals must have more than one form if they have an open form Minerals may have only one closed form Minerals may have only one closed form Mineral could have more than 1 form, closed or open Mineral could have more than 1 form, closed or open

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Open Form Open Form Prism Prism Requires additional forms Requires additional forms Closed Form Closed Form Cube Cube Does not require additional forms, but may contain them Does not require additional forms, but may contain them

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Example of multiple forms Cube {001}, octahedron {111}, and 3 prisms{110}, {101}, {011} Cube {001}, octahedron {111}, and 3 prisms{110}, {101}, {011} All forms have 4/m 3 2/m symmetry All forms have 4/m 3 2/m symmetry Two combined closed forms, plus 3 additional open forms {001} = cube {111} = octahedron Prisms {110} {101} {011} a c b

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Isometric forms 15 possible forms 15 possible forms 4 common ones 4 common ones Cube {001} – 4/m 3 2/m symmetry Cube {001} – 4/m 3 2/m symmetry Octahedron {111} – 4/m 3 2/m symmetry Octahedron {111} – 4/m 3 2/m symmetry Tetrahedron {111} – 4 symmetry Tetrahedron {111} – 4 symmetry Dodecahedron {110} Dodecahedron {110}

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Octahedron a c b c b a Both isometric forms: Tetrahedron {111} {111} Crystal class = 4/m 3 2/m Crystal class = 4 (111)

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Non-isometric form 10 types of forms 10 types of forms Pedion (open) Pedion (open) Single face Single face No symmetrically identical face No symmetrically identical face Pinacoid (open) Pinacoid (open) Two parallel faces Two parallel faces Related by mirror plane or inversion Related by mirror plane or inversion Dihedron (open - 2 types) Dihedron (open - 2 types) Two non-parallel face Two non-parallel face Related by mirror (dome) or 2-fold rotation (sphenoid) Related by mirror (dome) or 2-fold rotation (sphenoid)

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Fig Note: dome switches handedness Sphenoid retains handedness

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Prism (open) Prism (open) 3, 4, 6, 8 or 12 faces 3, 4, 6, 8 or 12 faces Intersect with mutually parallel edges forming a tube Intersect with mutually parallel edges forming a tube Pyramid (open) Pyramid (open) 3, 4, 6, 8, or 12 faces 3, 4, 6, 8, or 12 faces Intersect at a point Intersect at a point Dipyramid (closed) Dipyramid (closed) 6, 8, 12, 16, or 24 faces 6, 8, 12, 16, or 24 faces Two pyramids at each end of crystal Two pyramids at each end of crystal All of these forms are named on the basis of the shape of the cross section All of these forms are named on the basis of the shape of the cross section Total of 21 different forms Total of 21 different forms

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Fig Prisms Pyramids Dipyramids Cross section Open Closed Open Dihexagonal Hexagonal Ditrigonal Trigonal Ditetragonal TetragonalRhombic Three types – seven modifiers – total of 21 forms

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Trapezohedrons (closed) Trapezohedrons (closed) 6, 8, 12 faces 6, 8, 12 faces each a trapezoid (plane shape with 4 unequal sides) each a trapezoid (plane shape with 4 unequal sides) Named according to number of faces Named according to number of faces Scalenohedron (closed) Scalenohedron (closed) 8 or 12 faces 8 or 12 faces Each a scalene triangle (no two angles are equal) Each a scalene triangle (no two angles are equal)

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Rhombohedrons (closed) Rhombohedrons (closed) 6 faces, each rhomb shaped (4 equal sides, no 90 angles) 6 faces, each rhomb shaped (4 equal sides, no 90 angles) Looks like a stretched or shortened cube Looks like a stretched or shortened cube Tetrahedron (closed) Tetrahedron (closed) 4 triangular faces 4 triangular faces

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Fig. 2-33

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Combining forms Restrictions on types of forms within a crystal Restrictions on types of forms within a crystal All forms must be in the same crystal system All forms must be in the same crystal system All forms must have symmetry of one crystal class, for example: All forms must have symmetry of one crystal class, for example: Tetragonal prism has a single 4-fold rotation, only found in tetragonal crystal class with single 4-fold rotation axis Tetragonal prism has a single 4-fold rotation, only found in tetragonal crystal class with single 4-fold rotation axis Pedions never occur in mineral with center of symmetry Pedions never occur in mineral with center of symmetry

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Multi-faced forms are not composed of several simpler forms Multi-faced forms are not composed of several simpler forms A cube is not 6 pedions or 3 pinacoids A cube is not 6 pedions or 3 pinacoids

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Special relationships between forms Enantiomorphous forms Enantiomorphous forms Positive and negative forms Positive and negative forms

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Enantiomrophous Form Enantiomorphic forms contain a screw axis Enantiomorphic forms contain a screw axis Axis may rotate to the right or left Axis may rotate to the right or left The two forms generated are mirror images of each other The two forms generated are mirror images of each other

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Fig fold screw axis May be 2-fold, 4-fold, or 6-fold Atomic scale rotation Enantiomorphous forms result from either right or left spiral of screw axis Amino acids: Almost always left handed Through time convert to right handed Age-dating tool 0 < D/L < 1

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Enantiomorphous Forms Must lack center of symmetry and mirrors Must lack center of symmetry and mirrors Forms are related to each other by a mirror Forms are related to each other by a mirror right and left handed forms right and left handed forms Individual crystal either right or left handed, but not both Individual crystal either right or left handed, but not both Quartz is common example Quartz is common example

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Fig Crystal are mirror images of each other, but there are no mirror images in the crystals Enantiomorphous Forms

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Positive and Negative forms Created by rotation of a form Created by rotation of a form Rotation not present in the form itself Rotation not present in the form itself Two forms related to each other by mirror planes Two forms related to each other by mirror planes Mirror planes missing within the form itself Mirror planes missing within the form itself Two possible rotations: Two possible rotations: 60º on 3-fold rotation axis 60º on 3-fold rotation axis 90º on 4- or 2-fold rotation axis 90º on 4- or 2-fold rotation axis

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Fig Positive and negative faces in quartz crystal Quartz lacks center of symmetry Quartz Crystal

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Forms in the Six Crystal System Forms control orientation of crystallographic axes of the 6 crystal system Forms control orientation of crystallographic axes of the 6 crystal system Systematic relationship between form, symmetry present, and Hermann-Mauguin symbols Systematic relationship between form, symmetry present, and Hermann-Mauguin symbols Following slides show these relationships Following slides show these relationships

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Triclinic Common symmetry: 1-fold rotation Common symmetry: 1-fold rotation Table 2.2 Table 2.2 c-axis parallels prominent zone axis c-axis parallels prominent zone axis b and a axes parallel crystal edges b and a axes parallel crystal edges and typically > 90º and typically > 90º Single Hermann-Mauguin symbol Single Hermann-Mauguin symbol Common minerals: plagioclase and microcline Common minerals: plagioclase and microcline

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Fig Pedions Pinacoid 11 Triclinic a b c = zone axis

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Monoclinic Common symmetry: 2-fold rotation and/or single mirror plane Common symmetry: 2-fold rotation and/or single mirror plane b axis commonly parallel the 2-fold rotation and/or perpendicular to mirror plane b axis commonly parallel the 2-fold rotation and/or perpendicular to mirror plane c axis parallel to prominent zone c axis parallel to prominent zone a axis down and to front so > 90 a axis down and to front so > 90 Single H-M symbol (2, m, or 2/m) Single H-M symbol (2, m, or 2/m) Common minerals: amphiboles, pyroxenes, micas Common minerals: amphiboles, pyroxenes, micas

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Fig Monoclinic 2-fold rotation axis

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Orthorhombic Common symmetry: 3 2-fold rotations and/or 3 mirror planes Common symmetry: 3 2-fold rotations and/or 3 mirror planes Crystal axes are parallel to 2-fold rotations or perpendicular to mirror planes, or both Crystal axes are parallel to 2-fold rotations or perpendicular to mirror planes, or both Any axis could have any symmetry Any axis could have any symmetry Reported in H-M notation: Reported in H-M notation: 1 st = a axis, 2 nd = b axis, 3 rd = c axis 1 st = a axis, 2 nd = b axis, 3 rd = c axis E.g. mm2 – a mirror, b mirror, c parallel 2-fold rotation E.g. mm2 – a mirror, b mirror, c parallel 2-fold rotation

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Fig Orthorhombic a c b mm2 2/m2/m2/m 222 a a b b c c

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Tetragonal Common symmetry: single 4-fold rotation, or 4-fold rotoinversion Common symmetry: single 4-fold rotation, or 4-fold rotoinversion c axis always the single 4-fold rotation axis c axis always the single 4-fold rotation axis a and b coincide with 2-fold rotation or mirror (if present) a and b coincide with 2-fold rotation or mirror (if present) H-M symbol: H-M symbol: 1 st = c axis 1 st = c axis 2 nd = b and a axes 2 nd = b and a axes 3 rd = symmetry on [110] and [110] axis at 45º to a and b axes 3 rd = symmetry on [110] and [110] axis at 45º to a and b axes

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Example 42m 42m C = 4-fold rotoinversion C = 4-fold rotoinversion a and b axes [100] and [010] are 2-fold rotation a and b axes [100] and [010] are 2-fold rotation There are mirrors to [110] and [110] There are mirrors to [110] and [110]

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Fig m Positive and negative tetragonal tetrahedron a c b Note – tetragonal so a = b ≠ c, this is not an isometric form

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Hexagonal Common symmetry: 1 3-fold axis (trigonal division) or 1 6-fold axis (hexagonal division) Common symmetry: 1 3-fold axis (trigonal division) or 1 6-fold axis (hexagonal division) c axis parallel to 6-fold or 3-fold rotation c axis parallel to 6-fold or 3-fold rotation a axes parallel to 2-fold rotation or perpendicular to mirror a axes parallel to 2-fold rotation or perpendicular to mirror H-M symbols written with 1 st = c axis, 2 nd parallel a axes, 3 rd perpendicular to a H-M symbols written with 1 st = c axis, 2 nd parallel a axes, 3 rd perpendicular to a

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Figure 2-41 a1a1 -a 3 a2a2 c A prism and multiple dipyramids 6/m2/m2/m

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Isometric Common symmetry 4 3-fold axes Common symmetry 4 3-fold axes 3 equivalent symmetry axes coincide with crystallographic axes 3 equivalent symmetry axes coincide with crystallographic axes (e.g. for cube, it’s the 4 fold rotations) (e.g. for cube, it’s the 4 fold rotations) Symmetry either 2-fold or 2-fold Symmetry either 2-fold or 2-fold H-M symbols; H-M symbols; 1 st crystallographic axes 1 st crystallographic axes 2 nd diagonal axes [111] 2 nd diagonal axes [111] 3 rd center of one edge to center of another edge [110] 3 rd center of one edge to center of another edge [110]

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Fig /m32/m - Isometric 4/m 3 2/m a b c

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