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
TETRAGONAL
Drusy Quartz on Barite
HEXAGONAL
ORTHORHOMBIC
MONOCLINIC
TRICLINIC
Acknowledgement: the following images from Susan
and Stan Celestian,
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
the same letter “a”).
a2
They all meet at mutual 90o angles in the
center of the crystal.
a1
Crystals in this system are typically
blocky or ball-like.
ISOMETRIC
Basic Cube
ALL ISOMETRIC CRYSTALS HAVE 4
3-FOLD AXES
Crystal Systems

ISOMETRIC CRYSTALS
a3
a3
a2
a1
a2
a1
ISOMETRIC - Basic Cube
Fluorite cube with
crystal axes.
2
Crystal Systems

ISOMETRIC BASIC CRYSTAL SHAPES
Pyrite
Fluorite
Spinel
Octahedron
Cube
Garnet
Trapezohedron
Cube with
Pyritohedron
Striations
Garnet - Dodecahedron
Crystal Systems

HEXAGONAL CRYSTALS
c
HEXAGONAL - Three horizontal
axes meeting at angles of 120o and one
perpendicular axis.
a3
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:
subsytems:
1. Hexagonal
2. Trigonal
Crystal Systems

TETRAGONAL CRYSTALS
c
TETRAGONAL
Two equal, horizontal, mutually
perpendicular axes (a1, a2)
a1
a2
Vertical axis (c) is perpendicular to the
horizontal axes and is of a different length.
c
TETRAGONAL Crystal
Axes
a1
a2
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
a
b
ORTHORHMOBIC
Crystal Axes
a
b
An Alternative Crystal
Axes Orientation
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
unequal length.
c
Axes a and b are perpendicular.
b
a
MONOCLINIC Crystal
Axes
Axes b and c are perpendicular.
But a and c make some oblique
angle and with each other.
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
a
b
TRICLINIC Crystal
Axes
In this system, all of the axes are of
different lengths and none are
perpendicular to any of the others.
SYMMETRY: ONLY 1 OR 1-BAR
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
120o
120o
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
Triclinic
No Center
Center
1
1
Monoclinic
2, 2 (= m)
2/m
Orthorhombic
222, 2mm
2/m 2/m 2/m
4, 4, 422, 4mm, 42m
4/m, 4/m 2/m 2/m
Tetragonal
Hexagonal
Isometric
3, 32, 3m
3, 3 2/m
6, 6, 622, 6mm, 62m
6/m, 6/m 2/m 2/m
23, 432, 43m
2/m 3, 4/m 3 2/m
12
Six Crystal Systems
3-D Lattice Types
Name
axes
angles
a≠b≠c
a≠b≠c
α ≠ β ≠ γ ≠ 90o
α = γ = 90o β ≠ 90o
a≠b≠c
a1 = a2 ≠ c
α = β = γ = 90o
α = β = γ = 90o
a1 = a2 = a3 ≠ c
β = 90o γ = 120o
a1 = a2 = a3
α = β = γ ≠ 90o
α = β = γ = 90o
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Hexagonal
Hexagonal (4 axes)
Rhombohedral
a1 = a2 = a3
Isometric
+c
+a
β
γ
α
+b
Axial convention:
“right-hand rule”
rule”
c
c
c
b
a
b
b
P
P
a I
Monoclinic
α = γ = 90ο ≠ β
a≠b≠c
a
Triclinic
α ≠ β≠ γ
a≠b≠c
=C
c
a
b
P
C
F
Orthorhombic
α = β = γ = 90ο a ≠ b ≠ c
I
13
c
c
a2
a1
a1
a2
P
I
Tetragonal
α = β = γ = 90ο a1 = a2 ≠ c
P or C
R
Hexagonal
Rhombohedral
α = β = 90ο γ = 120ο α = β = γ ≠ 90ο
a1 = a2 = a3
a1 = a2 ≠ c
a3
a2
a1
P
F
I
Isometric
α = β = γ = 90 ο a1 = a 2 = 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 = 6fold in hexagonal, 4-fold in
tetragonal, and 3-fold in trigonal.
trigonal.
The three axes in isometric are 4 or
4bar. The b axis in monoclinic
crystals is a 2fold or m-normal.
fluorite
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”
angle” = the angle
between the faces
measured like this
120o
120o
120o
120o
120o
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
looking down c
b
a
b
a
c
Crystal Morphology
Given the following crystal:
b
a
How reference faces?
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
z
a
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”)
b
a
b
c
unknown face (y
(y)
reference face (x
(x)
1
2
1
1
∞
1
invert
2
1
1
1
1
∞
clear of fractions
2
1
0
x
y
Miller index of
face y using x as
the a-b reference face
(2 1 0)
a
21
What is the Miller Index of the reference face?
b
x
y
a
b
c
unknown face (x
(x)
reference face (x
(x)
1
1
1
1
∞
1
invert
1
1
1
1
1
∞
clear of fractions
1
1
0
(1 1 0)
Miller index of
the reference face
is always 1 - 1
(2 1 0)
a
What if we pick y as the reference. What is the MI of x?
b
x
y
a
b
c
unknown face (x
(x)
reference face (y
(y)
2
1
1
1
∞
1
invert
1
2
1
1
1
∞
clear of fractions
1
2
0
(1 2 0)
Miller index of
the reference face
is always 1 - 1
(1 1 0)
a
22
Which choice is correct?
1)
b
x = (1 1 0)
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
b
unit cell
shape if
y = (1 1 0)
b
unit cell
shape if
x = (1 1 0)
a
a
x
x
b
y
y
a
axial ratio = a/b = 0.80
a
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)
z
a
The Miller Indices of face z using x as the reference
b
w
(1 1 0)
(2 1 0)
a
b
c
unknown face (z)
reference face (x
(x)
1
1
∞
1
∞
1
invert
1
1
1
∞
1
∞
clear of fractions
1
0
0
(1 0 0)
a
z
Miller index of
face z using x (or
(or
any face)
face) as the
reference face
24
b
Can you index the rest?
(1 1 0)
(2 1 0)
(1 0 0)
a
b
(1 1 0)
(0 1 0)
(1 1 0)
(2 1 0)
(2 1 0)
(1 0 0)
a
(1 0 0)
(2 1 0)
(2 1 0)
(1 1 0)
(0 1 0)
(1 1 0)
25
3-D Miller Indices (an unusually complex example)
a
b
c
unknown face (XYZ
(XYZ))
reference face (ABC
(ABC))
2
1
2
4
2
3
invert
1
2
4
2
3
2
(1
4
3)
c
C
Z
clear of fractions
A
X
O
Miller index of
face XYZ using
ABC as the
reference face
Y
B
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}
210}
b
(1 1)
(0 1)
(1 1)
(2 1)
(2 1)
(1 0)
a
(1 0)
(2 1)
(2 1)
(1 1)
(0 1)
(1 1)
Form = a set of symmetrically equivalent faces
braces indicate a form {210}
210}
pinacoid
related by a mirror
or a 2-fold axis
prism
pyramid
dipryamid
related by n-fold
axis or mirrors
29
Form = a set of symmetrically equivalent faces
braces indicate a form {210}
210}
Quartz = 2 forms:
Hexagonal prism (m = 6)
Hexagonal dipyramid (m = 12)
Isometric forms include
Octahedron
Cube
_
111
111
__
111
_
111
halite
Dodecahedron
101
magnetite
011
_
110
110
_
101
garnet
_
011
30
All three combined:
001
011
101
_
111
111
_
110
010
100
__
111
_
101
110
_
111
_
011
Zone
Any group of faces || a common axis
Use of h k l as variables for a, b, c
intercepts
(h k 0)
0) = [001]
001]
If the MI’
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