Introduction to Mineralogy Dr. Tark Hamilton Chapter 6: Lecture Crystallography & External Symmetry of Minerals Camosun College GEOS 250 Lectures: 9:30-10:20 M T Th F300 Lab: 9:30-12:20 W F300

fig_06_14 Rotoinversion Inside a Sphere (Stereonet)

fig_06_15 2-, 3-, 4- & 6- Rotoinversion Projections 2-fold = m 3-fold = 3 + i 6-fold = 3 + m 4-fold = 4 + m

table_06_02 32 Bravais Lattices

table_06_03

fig_06_16 Tetragonal 422 & Hexagonal 622 Oblique perspective A 4, 4-A 2 ’s Equatorial Plane (Primitive circle) A 6, 6-A 2 ’s Oblique view Of symmetry axes 422 Equatorial Plane (Primitive circle) 422 Oblique view Of symmetry axes 622 β-High Quartz Phosgenite Pb 2 Cl 2 CO 3

fig_06_17 Stereoprojection normal to 3, II to 2 Original motif In lower hemisphere Diad-Rotated motif In upper hemisphere Motifs produced By triad normal to page Hexagonal: 1-A3, 3-A2 ‘s α-Quartz

fig_06_18 4-, 3- & 2-fold Symmetry Axes in a Cube Tetrads Connect Along Face Normals Diads Connect Along Edge Diagonals Triads connect Along body diagonals

fig_06_19 Rotational Axes Normal to Mirrors m lies along primitive Solid dot upper hemisphere

fig_06_20 Mirrors in the Tetragonal System Point groups preclude m’s Rotational Axes with Perpendicular m’s: Up & Down + Side by Side Tetrad Axis with Parallel m’s, Upper only

fig_06_21 Intersecting Mirror Planes: Reflected reflections = Rotations Orthorhombic Perspective Plan Views Orthorhombic Perspective Tetragonal Perspective Tetragonal Perspective

fig_06_22 NaCl Cube + Octahedron & Symmetry 54°44’

fig_06_23 32 Possible Point Groups & Symmetry

fig_06_24 Motifs & Stereonet Patterns for 32 point groups 3 Monoclinic patterns: 2 nd setting 7 Tetragonal Point Groups 2 Triclinic Point Groups 3 Orthorhombic Point Groups

fig_06_24cont Motifs & Stereonet Patterns Cont’d 12 Hexagonal Groups 5 Isometric Groups

table_06_04 What Symmetry element makes the center of symmetry appear? 1121

fig_06_25 Only 6 Different Crystal Systems Determined by Axial Lengths & Angles Triclinic a ≠ b ≠ c α ≠ β ≠ γ ≠ 90° Monoclinic a ≠ b ≠ c α = γ = 90°, β > 90° Orthorhombic a ≠ b ≠ c α = β = γ = 90° Tetragonal a = b ≠ c α = β = γ = 90° Isometric a = b = c α = β = γ = 90° Hexagonal a 1 = a 2 = a 120°, 90°

fig_06_26 Crystal Morphology & Crystallographic Axes c is Zone Axis a & b ~symmetric b is pole to β > 90° plane a b a 1 a 2 a 3 are poles to faces in equatorial zone & 4, 2 rotational axes Hey! Somebody Has to be Perfect!

table_06_05

fig_06_27 Orientation & Intercepts of Crystal Faces, Cleavages & Mirror Planes Intercepts at Integral Values of Unit Cell Edges Forms Correspond To Faces, Edges& Corners of Unit Cell

fig_06_28 Orthorhombic Crystal with 2 Pyramidal Forms Olivine 2/m2/m2/m (Similar forms in Scheelite 4m CaWO4)

Miller Indices Are integers derived from the intercepts on the a, b, & c axes Intercepts are expressed in terms of logical unit cell edge dimensions (the fundamental translation unit in the lattice) If a = 10.4 Å, then an intercept at 5.2 Å on the a axis is ½ Fractions are cleared by multiplying by a common denominator e.g. a plane cutting at [⅓ ⅔ 1/ ∞ ] X 3 = (1 2 0)

fig_06_29 Isometric Lattice, Intercepts & Miller Indices What would be the difference between crystals which had Cleavages or other planes along (100) versus (400)?

fig_06_30 Miller Indices for Positive & Negative Axes This Crystal like Diamond, Fluorite or Spinel has all Faces of the “form” (111)

A Crystal Form A Crystal Form is a group of Like crystal faces All faces of a given form have the same relationship to the symmetry of the crystal In Isometric Crystals the general form (100) includes: (010) (001) (-100) (0-10) and (00-1) through the 4-fold, 3-fold, 2-fold axes and Mirror Planes These faces will all tilt or intersect at 90° Triclinic forms: (100) (010) & (001) all have different pitches; so they do not belong to a single common form For 2 & 2/m Monoclinic forms (101) = (-10-1) ≠ (-101)

fig_06_31 Hexagonal 4-digit Miller-Bravais Indices a 1 a 2 a 3 c 1 Form : Prismatic (1010) (1100) (0110) (1010) (1100) (0110) 1 Form : Pyramidal (1121) (2111) (1211) (1121) (2111) (1211)

fig_06_32 Crystal Zones & Zone Axes Zone Axis [100] Zone : r’ c r b Zone Axis [001] Zone : m’ a m b Which Forms are : a)Prismatic b)Pyramidal? What is the General form Of the miller Index for : a)m b)r’ (hkl) = a single face [hkl] = a form or pole

fig_06_33 Conventional Lettering of Forms General Miller Indices For each form (hkl)? What symmetry makes p=p, m=m ?

fig_06_34 The (111) Form in 1 & 4/m 3 2/m Triclinic : Inversion Center Makes only (111) & (111) Isometric : Generates full Octahedron (111)

fig_06_35 Distinct Forms Manifest Different Details Striation patterns & directions differ For forms on Quartz Apophyllite KCa 4 (Si 4 O 10 ) 2 F – 8H 2 O 4/m 2/m 2/m Base : Pearly, others vitreous Striation patterns & directions differ for cube & pyritohedon forms on 2/m 3 Pyrite

table_06_06 15 Closed Forms Faces ≥ 4 18 Open Forms Faces ≤ 4

table_06_07 15 Closed Forms

fig_06_36a 11 Open Non-Isometric Forms & Symmetry 2 Dihedrons: 7 Prisms 1 Pedion & 1 Pinacoid 11 Open Forms Sphenoid = Angles Dome

fig_06_36b 14 Pyramidal Crystal Forms & Symmetry Pyramids: 7 Open Forms Dipyramids 7 Closed Forms Rhombic & Trigonal, Ditrigonal & etc. for both

fig_06_36c 8 Non & 8 Isometric Crystal Forms & Symmetry 3 Trapezohedrons (4≠angles, 4≠edges) 2 Scalenohedrons (3≠angles, 3≠edges) Rhombic equilateral 2 Disphenoids: Tetragonal isosceles 8 Isometric Forms 1 Rhombohedron (2 pairs=angles, 1 edge) 8 Non-isometric Forms Both Octahedrons & Tetrahedrons have Equilateral [111] forms Tristetrahedron, Trisoctahedron & Tertahexahedrons have isosceles triangle faces

fig_06_36d 7 Isometric Crystal Forms & Symmetry Dodecahedron & Deltoid 12 have Sym. Trapezoids Pyritohedron, Tetartoid & Gyroid (Pentagon faces)