Miller Indices

Lecture 4:
Miller Indices, Zones, Forms, Twins
Last lecture.. Crystal Systems and Crystallographic Axes were
introduced
These are used by mineralogists to describe the external form of
crystals…
The crystallographic axes define a COORDINATE SYSTEM.. If you
know what the coordinate system looks like, you can describe
the position of a plane (i.e. a crystal face) in space
There is a standard notation for describing the orientation of
crystal faces, cleavage planes and any other planar properties
of a mineral, called MILLER INDICES (h k l)
Read Perkins
Chpt 11

How to calculate a Miller Index:
1. Find axial intercepts
2. Invert axial intercepts (infinity => 0)
3. Clear fractions
Example:
c
1. Axial Intercepts:
a=1, b=2, c=2
2
2
b
2. Invert intercepts:
1/1, 1/2, 1/2
1
a
3. Clear fractions:
(2 1 1)

Example:
c
1. Axial Intercepts:
1
a=1, b=1, c=1
2. Invert intercepts:
1/1, 1/1, 1/1
1 1
a
b
3. Clear fractions:
(1 1 1)
c
1. Axial Intercepts:
a=1, b=1, c=infinity
2. Invert intercepts:
1/1, 1/1, 0
a
1 1
b
3. Clear fractions:
(1 1 0)

2
c
-a
1. Axial Intercepts:
a=-1, b=1, c=2
-1
2. Invert intercepts:
-1/1, 1/1, 1/2
a
1
b
3. Clear fractions:
(-2 2 1)

Law of Rational Indices (“Hauy’s Law”): Miller indices
can always be expressed by simple (small) whole numbers
or zero
Zone [u v w]: a direction defined by a set of crystal faces
with parallel intersection edges; the zone axis is a line through the
center of the crystal that is parallel to the lines of face
intersections
To calculate, choose 2 faces (h 1 k 1 l 1 ) and (h 2 k 2 l 2 )
Example (100) and (10-1)
Write h 1 k 1 l 1 h 1 k 1 l 1 1 0 0 1 0 0
h 2 k 2 l 2 h 2 k 2 l 2 1 0-1 1 0 -1
[uvw]=(k 1 l 2 -l 1 k 2 )(l 1 h 1 -h 1 l 2 )(h 1 k 2 -k 1 h 2 ) =
(0-0), (0+1), (0-0) = [010]

FORM {h k l}: a group of crystal faces, related by
symmetry
So what? It means those faces all have the same chemical
and physical properties, because they have the same
underlying atomic structure
Note: Don’t confuse FORM with HABIT
HABIT = the external shape of a crystal specimen depends
on symmetry, number and size of forms present, depends
on growth conditions

The number of faces that belong to a FORM is determined
by the symmetry of the point group (Ex. Figs 5.36, 5.37 in
Klein)
Ex. Bar 1
Only have a center of symmetry, so form is just two
parallel faces (“PINACOID”)
Ex. 4/mbar32/m
One face (1 1 1) will get reflected and rotated to form
seven more faces (“OCTAHEDRON”)

More about forms…
GENERAL FORMS: the faces in this form intersect the
crystallographic axes at different lengths
SPECIAL FORMS: all other forms; these faces are parallel
or perpendicular to any of the symmetry elements in the
crystal class

More about forms…
CLOSED FORM: encloses space without the presence of
other forms
Ex. Dipyramid: 2 pyramids related by a mirror
Ex. Cube: 6 square faces
Ex. Octahedron: 8 equilateral triangle shaped faces
(tetragonal dipyramid)
OPEN FORM: does not completely enclose space
Ex. Pedion: single face
Ex. Pinacoid: two parallel faces
There are 48 possible forms, which can occur in infinitely many sizes.

TWIN: a symmetrical intergrowth of two or more crystals of
the same substance
Twin element: new symmetry element that relates one part
of the twin aggregate to the other part, they include.. TWIN
PLANE, TWIN AXIS, TWIN CENTER
Lots of kinds of twins, including:
Contact Twins: two crystals appear to be touching; share a plane of
atoms
Penetration Twins: two crystal look like they’ve grown through each other;
share a volume of atoms
Multiple Twins: 3 or more parts of the crystal are twinned according to the
same twin law; most common are polysynthetic twins