Indexing a Plane

This is a 3D array of atoms in a primitive cubic lattice.

It can be represented as periodic unit cells.

Now consider a single unit cell.

We can define a set of axes, x, y and z, that describe the edges of this unit cell.

The lengths of the edges are defined by distances, known as the lattice parameters:

a on the x-axis,

  b on the y-axis,

    c on the z-axis.

Suppose we have a plane within this unit cell.

This plane intercepts:

  the x-axis at a/2,

    the y-axis at b/1,

      the z-axis at c/2,

Taking reciprocals of these intercepts gives the Miller indices:

   (212)

If the intercepts are:

1/h on the x-axis,
  1/k on the y-axis,
    1/l on the z-axis.

The Miller indices are (hkl)

Or, if the resulting indices contain fractions; multiply throughout to give integer values of h,k, and l.