Camosun College GEOS 250
Lectures: 9:30-10:20 M T Th F300
Lab: 9:30-12:20 W F300

fig_06_14

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

table_06_02

table_06_03

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

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

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

Solid dot upper hemisphere m lies along primitive fig_06_19

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

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

NaCl Cube + Octahedron & Symmetry
54 °44’ fig_06_22

32 Possible Point Groups & Symmetry fig_06_23

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

5 Isometric Groups
12 Hexagonal Groups fig_06_24cont

21 11
What Symmetry element makes the center of symmetry appear?
table_06_04

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°
Hexagonal a
1
= a
2
= a
3
@ 120 °, c@ 90°
Isometric a = b = c
α = β = γ = 90° fig_06_25

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!
fig_06_26

table_06_05

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_27

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

• 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)

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

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

• 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)

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_31

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

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

Triclinic :
Inversion Center
Makes only
(111) & (111)
Isometric :
Generates full
Octahedron (111) fig_06_34

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

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

15 Closed Forms table_06_07

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

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

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

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