Miller Indices are used in crystallography, to describe the
orientation of a plane, or a set of parallel planes of atoms in a
crystal.

To appreciate the usefulness of Miller Indices, we first have to
understand what a crystal is. A crystal is a solid that consists of
atoms arranged in a periodic pattern. Gases and
liquids are by definition not crystalline. However, not all
solids are crystalline either; some are amorphous.

When doing calculations on crystals, it is convenient to ignore the
actual atoms, their atomic radius, and think of the crystal as a set
of imaginary points, with a fixed relation in space (similar to a
wire-frame). Imagine this wireframe to be infinitely long in all three
dimensions; *real* crystals of course have a finite size, but
they generally consist of a *very large* number of atoms in all
directions (the crystal lattice).

The crystal lattice can be described by a regular ordering of *unit cells*. Three vectors **a**, **b**, and **c**
determine the size and shape of the unit cell, forming a
parallelepiped. If **a**, **b**, and **c** are equal in
length *and* the angles between the vectors are 90°, the
shape is a special kind of parallelepiped: a cube, but this is not
always the case.

Now we want to describe planes in the crystallographic lattice.
Due to the similarity and repetition of the unit cell we don't need to
describe *each* plane by its atom positions, but we can formulate
a set of indices describing similar planes (planes with the same
direction and spacing). These indices are called the *Miller
Indices*, named after the English crystallographer William H.
Miller (1801-1880). The Miller Indices are calculated as follows:

- Determine the intercepts of the plane along the
crystallographic axes
**a**, **b**, and **c**, in terms of the
dimensions of the unit cell
- Take the reciprocals of the intercept values
- Clear the fractions (do
*not* multiply by a negative factor)
- Reduce to the lowest terms.

If the axis intercepts are 2, 1, and 3, the Miller indices are calculated by:

The reciprocals: 1/2, 1/1, 1/3
Clear the fractions (multiply by 6): 3, 6, 2
Reduce to the lowest terms (already done).
Thus, the Miller indices are 3, 6, and 2, which is denoted by (362).
Negative intercepts are denoted with a bar over the corresponding
index. The Miller indices usually refer to the plane closest to the
origin, but they can also taken as referring to any other plane in the
set with the same direction, or the whole set of planes taken together.

Planes that run parallel along an axis have an intercept at infinity,
and the Miller Index is zero. This is shown for a cubic cell:

___________
/| /|
/ | / |
/__|_______/ |
| | | |
c | |_______|__|
| | / | /
| b | / | /
| / |/_________|/ the unit cell
| /
|/_________a
O
/| /|
/ | / |
/ | / |
| | | |
c | | | |
| | / | /
| b | / | /
| / |/ |/ (100)
| /
|/_________a
O
/| /| /|
/ | / | / |
/ | / | / |
| | | | | |
c | | | | | |
| | / | / | /
| b | / | / | /
| / |/ |/ |/ (200)
| /
|/_________a
O
___________
/ /
/ /
/_________ /
c __________
| / /
| b / /
| / /_________ / (001)
| /
|/_________a
O

The system of Miller Indices describes *relative* plane
spacings and directions. Compare this with the walls in a building. All
the floors (considering that they are equally spaced) can be described
by Miller Indices. All the *even* floors could also be described
with Miller Indices, with the same direction, but different magnitude.
All the north-walls (considering the rooms are equally big) have a
different set of Miller Indices.

**Now what's the point of all this?**

Crystallographers describe planes because they have different
properties. This can be easily (as much as ASCII allows this) seen in a
two-dimensional plot of a crystal lattice (given by the vectors **a**
and **b**).

b
O__. . . . . . .
a | /// | | |
. ./// . . | | |
/// | | |
. /// . . . . . .
///
. . . . . . . .

Shown are the planes with Miller indices (23) and (01) (left, and right
respectively). Planes with *lower* Miller Indices have a
*greater* spacing and a *higher* density of lattice points
(run trough more atoms in the lattice per unit length).

It is these properties of the crystallographic planes that allow us
to characterize a crystal using X-ray diffraction. These properties
also greatly determine the physical properties of the crystal. For instance: think about
cutting a diamond along different planes. Some directions are easier to
cut than other directions. The Miller Indices are thus a convenient way to
characterize crystal structures.