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It is very easy to visualize the location of a simple face given miller index, or to derive a miller index from simple faces c -b b a -c Fig. 2-23

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**Hexagonal Miller index**

There need to be 4 intercepts (hkil) h = a1 k = a2 i = a3 l = c Two a axes have to have opposite sign of other axis so that h + k + i = 0 Possible to report the index two ways: (hkil) (hkl)

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(1120) (1121) (1010) (100) (110) (111) Klein and Hurlbut Fig. 2-33

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**Assigning Miller indices**

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

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

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

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

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Drusy quartz Globular hematite

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**Terminology useful for describing general shapes of minerals**

(asbestos: amphiboles and pyroxenes) (table-like) (knife like – kyanite) (Mica) Fig. 2-47

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**Fibrous tremolite: amphibole Ca(Mg,Fe)5Si8O22 (OH)2**

Bladed kyanite Al2SiO5

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**Zones Collection of common faces Parallel to some common line**

Line called the zone axis Identified by index [hkl] Zone axis parallels intersection of edges of faces

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**Zone axis intesects (001) lattice nodes = [001]**

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

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**Other linear crystallographic directions**

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

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**Form Formal crystallographic nomenclature of the shape of minerals**

Description Collection of crystal faces Related to each other by symmetry Identified by index: {hkl} 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):**

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

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**Possible to determine the shape of a form with:**

Miller index of one face in form Point symmetry of the crystal class The form is created by operating point symmetry on the initial face Number of faces in a form depends on crystal class

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**– called a rhombic prism Rhombus – an equilateral parallelogram **

Face parallel to a axis Mirror parallel to (010) Mirror parallel to (001) {011} form in crystal class with point symmetry 2/m 2/m 2/m (Orthorhombic) – called a rhombic prism Rhombus – an equilateral parallelogram Prism – a crystal form whose faces are parallel to one axis Fig. 2-29

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**Triclinic system: Point group (i.e. crystal class) = 1**

Symmetry content = (1A1) {111} has only 2 faces

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**Isometric system: Point group (crystal class) = 4/m 3 2/m**

Symmetry content = 3A4, 4A3, 6A2, 9m {111} has 8 faces Form is an octahedron

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**Isometric system Point group (crystal class) = 4**

Symmetry content = 1A4 {111} has 4 faces Form is a tetrahedron (111) C b a

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**Minerals must have more than one form if they have an open form **

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

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**Open Form Closed Form Prism Requires additional forms Cube**

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

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**Isometric forms 15 possible forms 4 common ones**

Cube {001} – 4/m 3 2/m symmetry Octahedron {111} – 4/m 3 2/m symmetry Tetrahedron {111} – 4 symmetry Dodecahedron {110}

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**Both isometric forms: {111} {111} Tetrahedron Octahedron**

(111) b a b a Tetrahedron Octahedron {111} {111} Crystal class = 4 Crystal class = 4/m 3 2/m

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**Non-isometric form 10 types of forms Pedion (open) Pinacoid (open)**

Single face No symmetrically identical face Pinacoid (open) Two parallel faces Related by mirror plane or inversion Dihedron (open - 2 types) Two non-parallel face Related by mirror (dome) or 2-fold rotation (sphenoid)

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**Note: dome switches handedness Sphenoid retains handedness**

Fig. 2-31

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**Prism (open) Pyramid (open) Dipyramid (closed)**

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

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**Three types – seven modifiers – total of 21 forms**

Prisms Open Pyramids Open Closed Dipyramids Cross section Rhombic Tetragonal Trigonal Hexagonal Ditetragonal Ditrigonal Dihexagonal Fig. 2-32

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**Trapezohedrons (closed)**

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

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**Rhombohedrons (closed)**

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

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Fig. 2-33

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**Combining forms Restrictions on types of forms within a crystal**

All forms must be in the same crystal system 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 Pedions never occur in mineral with center of symmetry

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**Multi-faced forms are not composed of several simpler forms**

A cube is not 6 pedions or 3 pinacoids

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**Special relationships between forms**

Enantiomorphous forms Positive and negative forms

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**Enantiomrophous Form Enantiomorphic forms contain a screw axis**

Axis may rotate to the right or left The two forms generated are mirror images of each other

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**Almost always left handed Through time convert to right handed **

3-fold screw axis 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 May be 2-fold, 4-fold, or 6-fold Fig. 2.20

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**Enantiomorphous Forms**

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

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**Enantiomorphous Forms**

Crystal are mirror images of each other, but there are no mirror images in the crystals Fig. 2-34

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**Positive and Negative forms**

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

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**Positive and negative faces in quartz crystal**

Quartz lacks center of symmetry Quartz Crystal Fig. 2-35

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**Forms in the Six Crystal System**

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

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**Triclinic Common symmetry: 1-fold rotation**

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

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Triclinic c = zone axis Pedions Pinacoid b a 1 1 Fig. 2-36

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**Monoclinic Common symmetry: 2-fold rotation and/or single mirror plane**

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

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Monoclinic 2-fold rotation axis Fig. 2-37

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Orthorhombic 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 Any axis could have any symmetry Reported in H-M notation: 1st = a axis, 2nd = b axis, 3rd = c axis E.g. mm2 – a ^ mirror, b ^ mirror, c parallel 2-fold rotation

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Orthorhombic c c c b b 222 a b a a 2/m2/m2/m mm2 Fig. 2-38

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

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**Example 42m C = 4-fold rotoinversion**

a and b axes [100] and [010] are 2-fold rotation There are mirrors ^ to [110] and [110]

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c 42m Positive and negative tetragonal tetrahedron Note – tetragonal so a = b ≠ c, this is not an isometric form b a Fig. 2-39

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Hexagonal 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 a axes parallel to 2-fold rotation or perpendicular to mirror H-M symbols written with 1st = c axis, 2nd parallel a axes, 3rd perpendicular to a

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c A prism and multiple dipyramids a2 -a3 a1 6/m2/m2/m Figure 2-41

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**Isometric Common symmetry 4 3-fold axes**

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

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4/m32/m - Isometric 3 c b a 2/m 4/m Fig. 2-44

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Lecture 11 Crystallography

Lecture 11 Crystallography

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