Crystal Axes, Systems, Mineral Face Notation (Miller Indices)

A CRYSTAL is the outward form of the internal structure of the mineral. The 6 basic crystal systems are:

ISOMETRIC HEXAGONAL TETRAGONAL ORTHORHOMBIC Drusy Quartz on Barite MONOCLINIC TRICLINIC

Acknowledgement: the following images from Susan and Stan Celestian, Glendale Community College

Crystal Systems

CRYSTAL SYSTEMS are divided into 6 main groups.

The first group is the ISOMETRIC. This literally means “equal measure” and refers to the equal size of the crystal axes.

ISOMETRIC - Fluorite Crystals

1 Crystal Systems

 ISOMETRIC CRYSTALS

a3 ISOMETRIC In this crystal system there are 3 axes. Each has the same length (as indicated by a2 the same letter “a”). They all meet at mutual 90o angles in the a1 center of the crystal. Crystals in this system are typically ISOMETRIC blocky or ball-like. Basic Cube ALL ISOMETRIC CRYSTALS HAVE 4 3-FOLD AXES

Crystal Systems

 ISOMETRIC CRYSTALS

a3 a3

a1 a2 a2 a1 Fluorite cube with crystal axes. ISOMETRIC - Basic Cube

2 Crystal Systems

 ISOMETRIC BASIC CRYSTAL SHAPES

Spinel Fluorite Pyrite Octahedron Cube Cube with Pyritohedron Striations

Garnet Garnet - Dodecahedron Trapezohedron

Crystal Systems

 HEXAGONAL CRYSTALS

c

HEXAGONAL - Three horizontal axes meeting at angles of 120o and one a3 perpendicular axis. a2

a1 ALL HEXAGONAL CRYSTALS HAVE A SINGLE 3- OR 6-FOLD AXIS = C

HEXAGONAL Crystal Axes

3 Crystal Systems

 HEXAGONAL CRYSTALS

These hexagonal CALCITE crystals nicely show the six sided prisms as well as the basal pinacoid.

Two subsytems: 1. Hexagonal 2. Trigonal

Crystal Systems

 TETRAGONAL CRYSTALS

c TETRAGONAL Two equal, horizontal, mutually perpendicular axes (a1, a2) Vertical axis (c) is perpendicular to the a2 a1 horizontal axes and is of a different length. c

a2 TETRAGONAL Crystal a1 Axes

This is an Alternative Crystal Axes

4 Crystal Systems

 TETRAGONAL CRYSTALS

Same crystal seen edge on. WULFENITE

ALL HEXAGONAL CRYSTALS HAVE A SINGLE 3- OR 6-FOLD AXIS = C

Crystal Systems

 ORTHORHOMBIC CRYSTALS

ORTHORHOMBIC Three mutually perpendicular axes of different lengths. c c b a b a

An Alternative Crystal Axes Orientation ORTHORHMOBIC Crystal Axes Each axis has symmetry, either 2-fold or m-normal

5 Crystal Systems

 ORTHORHOMBIC CRYSTALS

Topaz from Topaz Mountain, Utah.

Crystal Systems

 ORTHORHOMBIC CRYSTALS

BARITE is also orthorhombic. The view above is looking down the “c” axis of the crystal.

6 Crystal Systems

 ORTHORHOMBIC CRYSTALS

Pinacoid View

Prism View STAUROLITE

Crystal Systems

 MONOCLINIC CRYSTALS

MONOCLINIC In this crystal form the axes are of c unequal length. Axes a and b are perpendicular. b Axes b and c are perpendicular. a But a and c make some oblique angle and with each other. MONOCLINIC Crystal Axes MONO = ONE AXIS OF SYMMETRY (2-FOLD OR MIRROR) = TO “b”

7 Crystal Systems

 MONOCLINIC CRYSTALS

Top View

Gypsum Mica

Orthoclase

Crystal Systems

 TRICLINIC CRYSTALS

c TRICLINIC In this system, all of the axes are of different lengths and none are a b perpendicular to any of the others.

SYMMETRY: ONLY 1 OR 1-BAR TRICLINIC Crystal Axes

8 Crystal Systems

 TRICLINIC CRYSTALS

Microcline, variety Amazonite

Crystal Faces Remember: sphene

Space groups for atom symmetry

Point groups for crystal face symmetry

Crystal Faces = limiting surfaces of growth Depends in part on shape of building units & physical conditions (T, P, matrix, nature & flow direction of solutions)

Acknowledgement: the following images from John Winter, Whitman College, WA

9 Crystal Morphology

Observation: The frequency with which a given face in a crystal is observed is proportional to the density of lattice nodes along that plane

Crystal Morphology Observation: The frequency with which a given face in a crystal is observed is proportional to the density of lattice nodes along that plane

10 Crystal Morphology Because faces have direct relationship to the internal structure, they must have a direct and consistent angular relationship to each other

Corundum, var. ruby

Crystal Morphology Nicholas Steno (1669): Law of Constancy of Interfacial Angles

120o 120o 120o Quartz o 120o 120

120o 120o

11 Crystal Morphology Diff planes have diff atomic environments

Crystal Morphology Crystal symmetry conforms to 32 point groups → 32 crystal classes in 6 crystal systems Crystal faces have symmetry about the center of the crystal so the point groups and the crystal classes are the same

Crystal System No Center Center Triclinic 1 1 Monoclinic 2, 2 (= m) 2/m Orthorhombic 222, 2mm 2/m 2/m 2/m Tetragonal 4, 4, 422, 4mm, 42m 4/m, 4/m 2/m 2/m Hexagonal 3, 32, 3m 3, 3 2/m 6, 6, 622, 6mm, 62m 6/m, 6/m 2/m 2/m Isometric 23, 432, 43m 2/m 3, 4/m 3 2/m

12 Six Crystal Systems

3-D Lattice Types Name axes angles o +c Triclinic a ≠ b ≠ c α ≠ β ≠ γ ≠ 90 Monoclinic a ≠ b ≠ c α = γ = 90o β ≠ 90o Orthorhombic a ≠ b ≠ c α = β = γ = 90o + o a Tetragonal a1 = a2 ≠ c α = β = γ = 90 β Hexagonal ≠ β o γ = o Hexagonal (4 axes) a1 = a2 = a3 c = 90 120 γ o α Rhombohedral a1 = a2 = a3 α = β = γ ≠ 90 o Isometric a1 = a2 = a3 α = β = γ = 90 +b Axial convention: “right-hand rule”

c c c

b b b a P a P a I = C Triclinic Monoclinic α ≠ β ≠ γ α = γ = 90ο ≠ β ≠ ≠ a b c a ≠ b ≠ c c

b a P C F I Orthorhombic α = β = γ = 90ο a ≠ b ≠ c

13 c c

a2

a a1 a 2 P or C R 1 P I Hexagonal Rhombohedral Tetragonal α = β = 90ο γ = 120ο α = β = γ ≠ 90ο α = β = γ = 90ο a = a ≠ c = ≠ 1 2 a1 a2 c a1 = a2 = a3

a3

a2 a 1 P F I Isometric α = β = γ = 90ο a1 = a2 = a3

Triclinic and Monoclinic

14 Orthorhombic and Tetragonal

Hexagonal

15 Isometric

Crystal Morphology Crystal Axes: generally taken as parallel to the edges (intersections) of prominent crystal faces

b

a

c halite

16 Crystal Morphology Crystal Axes: Symmetry also has a role: c = 6- fold in hexagonal, 4-fold in fluorite tetragonal, and 3-fold in trigonal. The three axes in isometric are 4 or 4bar. The b axis in monoclinic crystals is a 2fold or m-normal.

The crystallographic axes determined by x-ray and by the face method nearly always coincide. This is not coincidence!!

Crystal Morphology How do we keep track of the faces of a crystal?

17 Crystal Morphology How do we keep track of the faces of a crystal? Remember, face sizes may vary, but angles can't

Note: “interfacial angle” = the angle 120o 120o 120o between the faces between the faces o o 120 measured like this 120

120o 120o

Miller Indices of Crystal Faces How do we keep track of the faces of a crystal? Remember, face sizes may vary, but angles can't Thus it's the orientation & angles that are the best source of our indexing

Miller Index is the accepted indexing method It uses the relative intercepts of the face in question with the crystal axes

18 Crystal Morphology Given the following crystal:

2-D view b looking down c

a b a

c

Crystal Morphology Given the following crystal:

b How reference faces? a a face? b face? -a and -b faces?

19 Crystal Morphology Suppose we get another crystal of the same mineral with 2 other sets of faces: b How do we reference them? w x y

b a a z

Miller Index uses the relative intercepts of the faces with the axes Pick a reference face that intersects both axes Which one?

b b

w x x y

y

a a z

20 Which one? Either x or y. The choice is arbitrary. Just pick one. Suppose we pick x

b b

w x x y

y

a a z

MI process is very structured (“cook book”)

a b c

unknown face (y) 1 1 ∞ reference face (x) 2 1 1 b

invert 2 1 1 1 1 ∞ clear of fractions 2 1 0 x Miller index of y face y using x as (2 1 0) the a-b reference face a

21 What is the Miller Index of the reference face?

a b c

unknown face (x) 1 1 ∞ reference face (x) 1 1 1 b

invert 1 1 1 1 1 ∞ clear of fractions 1 1 0 x (1 1 0) Miller index of y the reference face (2 1 0) is always 1 - 1 a

What if we pick y as the reference. What is the MI of x?

a b c

unknown face (x) 2 1 ∞ reference face (y) 1 1 1 b

invert 1 1 1 2 1 ∞ clear of fractions 1 2 0 x (1 2 0) Miller index of y the reference face (1 1 0) is always 1 - 1 a

22 Which choice is correct?

1) x = (1 1 0) b y = (2 1 0) 2) x = (1 2 0) y = (1 1 0) x The choice is arbitrary y What is the difference? a

What is the difference?

b unit cell unit cell b shape if b shape if y = (1 1 0) x = (1 1 0)

a a x x

b y y

a a

axial ratio = a/b = 0.80 axial ratio = a/b = 1.60

23 What are the Miller Indices of all the faces if we choose x as the reference? Face Z?

b w (1 1 0) (2 1 0)

a z

The Miller Indices of face z using x as the reference

a b c

unknown face (z) 1 ∞ ∞ reference face (x) 1 1 1

b invert 1 1 1 1 ∞ ∞ w (1 1 0) clear of fractions (2 1 0) 1 0 0

(1 0 0) Miller index of a z face z using x (or any face) as the reference face

24 b Can you index the rest?

(1 1 0) (2 1 0)

(1 0 0) a

b

(0 1 0) (1 1 0) (1 1 0) (2 1 0) (2 1 0)

(1 0 0) (1 0 0) a

(2 1 0) (2 1 0)

(1 1 0) (1 1 0) (0 1 0)

25 3-D Miller Indices (an unusually complex example) a b c c unknown face (XYZ) 2 2 2 reference face (ABC) 1 4 3

C invert 1 4 3 2 2 2 Z clear of fractions (1 4 3)

O A Miller index of X Y face XYZ using ABC as the B reference face a b

Example: the (110) surface

26 Example: the (111) surface

Example: (100), (010), and (001)

(001)

(010)

For the isometric system, these faces are equivalent (=zone)

27 Miller Indices - 3 axes

Miller - Bravais Indices

Hexagonal System

28 Form = a set of symmetrically equivalent faces braces indicate a form {210}

b (0 1) (1 1) (1 1) (2 1) (2 1)

(1 0) (1 0) a

(2 1) (2 1) (1 1) (1 1) (0 1)

Form = a set of symmetrically equivalent faces braces indicate a form {210}

pinacoid prism pyramid dipryamid

related by a mirror related by n-fold or a 2-fold axis axis or mirrors

29 Form = a set of symmetrically equivalent faces braces indicate a form {210}

Quartz = 2 forms: Hexagonal prism (m = 6) Hexagonal dipyramid (m = 12)

Isometric forms include Octahedron Cube _ 111 111

__ _ 111 111

halite Dodecahedron

magnetite

011 101

_

110 110 garnet _ _ 101 011

30 All three combined:

001

011 _ 101 111 111

_ 110 010 100 110

__ 111 _ 111 _ _ 011 101

Zone Any group of faces || a common axis Use of h k l as variables for a, b, c intercepts (h k 0) = [001] If the MI’s of 2 non-parallel faces are added, the result = MI of a face between them & in the same zone

31 Isometric Forms

Pyrite

The Cubic Form of Native Copper

32 Stereographic Projections

Stereographic Projections

Illustrated above are the stereographic projections for Triclinic point groups 1 and -1

33 The 32 Point Groups

The Triclinic System microcline

34 The Monoclinic System

Orthoclase

The Orthorhombic System barite

anhydrite

35 The Trigonal Subsystem

The Trigonal Subsystem

Tourmaline on quartz

36 The Tetragonal System

vesuvianite scapolite

The Tetragonal System

rutile

37 topaz The Tetragonal System

The Hexagonal System

cinnabar

38 beryl The Hexagonal System

The Hexagonal System

zincite

39 garnet

The Isometric System

The Isometric System

pyrit e

40