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)