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Introduction to MineralogyDr. Tark Hamilton
Chapter 6: Lecture 23-26Crystallography & 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 perspective422
1-A4 , 4-A2’s
Equatorial Plane(Primitive circle)
6221-A6 , 6-A2’s
Oblique viewOf symmetry axes
422
Equatorial Plane(Primitive circle)
422
Oblique viewOf symmetry axes
622
β-High Quartz
Phosgenite Pb2Cl2CO3
fig_06_17
Stereoprojection normal to 3, II to 2
Original motifIn lower hemisphere
Diad-Rotated motifIn upper hemisphere
Motifs producedBy 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 ConnectAlong Face Normals
Diads ConnectAlong Edge Diagonals
Triads connectAlong body diagonals
fig_06_19
Rotational Axes Normal to Mirrors
m lies along primitive
Solid dot upperhemisphere
fig_06_20
Mirrors in the Tetragonal System
Point groups preclude m’s
Rotational Axeswith Perpendicular m’s:
Up & Down + Side by Side
Tetrad Axis withParallel m’s, Upper only
fig_06_21
Intersecting Mirror Planes:Reflected reflections = Rotations
OrthorhombicPerspective
Plan Views
OrthorhombicPerspective
TetragonalPerspective
TetragonalPerspective
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: 2nd setting
7 TetragonalPoint Groups
2 TriclinicPoint Groups
3 OrthorhombicPoint Groups
fig_06_24cont
Motifs & Stereonet Patterns Cont’d
12 Hexagonal Groups 5 Isometric Groups
table_06_04What Symmetry element makes the center of symmetry appear?
1121
fig_06_25
Only 6 Different Crystal SystemsDetermined by Axial Lengths & Angles
Triclinica ≠ b ≠ c
α ≠ β ≠ γ ≠ 90°
Monoclinica ≠ b ≠ c
α = γ = 90°, β > 90°
Orthorhombica ≠ b ≠ c
α = β = γ = 90°
Tetragonala = b ≠ c
α = β = γ = 90°
Isometrica = b = c
α = β = γ = 90°
Hexagonala1 = a2 = a3
@ 120°, c@ 90°
fig_06_26
Crystal Morphology & Crystallographic Axes
c is Zone Axisa & b ~symmetric
b is pole to β > 90° plane
a b a1 a2 a3
are poles to facesin equatorial zone
& 4, 2 rotational axes Hey! SomebodyHas to be Perfect!
table_06_05
fig_06_27
Orientation & Intercepts of Crystal Faces, Cleavages & Mirror Planes
Intercepts at Integral Valuesof Unit Cell Edges
Forms CorrespondTo 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 hadCleavages or other planes along (100) versus (400)?
fig_06_30
Miller Indices for Positive & Negative Axes
This Crystal like Diamond, Fluorite or Spinel has allFaces 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
a1 a2 a3 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 theGeneral formOf the millerIndex for :
a) mb) r’
(hkl) = a single face[hkl] = a form or pole
fig_06_33
Conventional Lettering of FormsGeneral Miller IndicesFor each form (hkl)?
What symmetrymakes 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 KCa4(Si4O10)2 F – 8H2O
4/m 2/m 2/mBase : Pearly, others vitreous
Striation patterns &directions differ for cube & pyritohedon
forms on 2/m 3 Pyrite
table_06_06
15 Closed FormsFaces ≥ 4
18 Open FormsFaces ≤ 4
table_06_07
15 Closed Forms
fig_06_36a
11 Open Non-Isometric Forms & Symmetry2 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
Dipyramids7 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 isoscelestriangle faces
fig_06_36d
7 Isometric Crystal Forms & Symmetry
Dodecahedron &Deltoid 12 have
Sym. Trapezoids
Pyritohedron,Tetartoid & Gyroid(Pentagon faces)