Miller indices

Definition
Miller indices are integer sets that were created to distinguish directions and planes in a lattice. They are used primarily in crystalline structures because they describe planes in relation to a larger lattice in relative terms, as opposed to absolute terms. An example of this is describing planes in a building, Miller indices would distinguish the floor from the walls, and north wall from west wall, however it would not distinguish the 4th floor from the 5th floor. This is useful in crystal lattices because the planes are the same in many directions (like floors in a tall building).

Calculating Miller Indices
To calculate Miller indices take the reciprocal of the x, y, z intercepts and reduce the result down to lowest integers. If the xyz intercepts of a plane are (1,0,2) the reciprocal is (1,0,1/2) and the Miller Indices are 2 0 1.

Negative intercepts can be shown by placing a bar over the corresponding index. Common notation is to put the indices in brackets when they describe the form of a crystal and to leave them without brackets if they describe a face of the crystal. Named after British mineralogist William H. Miller.

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:

1. Determine the intercepts of the plane along the crystallographic axes a, b, and c, in terms of the dimensions of the unit cell
2. Take the reciprocals of the intercept values
3. Clear the fractions (do not multiply by a negative factor)
4. 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.