6 - Lecture notes for GEOL3010

Crystallography: Miller Indices

Recall our discussion about the likelihood of developing a particular crystal face is, in part, related to the density of lattice nodes (and in part to the rate at which a crystal grows).

Many times we wish to discuss a particular crystal face or more importantly, a particular plane of atoms within the crystal lattice structure. To do this, a universally accepted system of indices has been developed to describe the orientation of crystallographic planes and crystal faces relative to crystallographic axes. This convention is called the system of Miller indices.

Miller Indices are a symbolic vector representation for the orientation of an atomic plane in a crystal lattice and are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes.

The method by which indices are determined is best shown by example. Recall, that there are three axes in crystallographic systems (*sometimes in the hexagonal system adopts a convention where there are four axes). Miller indices are represented by a set of 3 integer numbers.

Example of the (111) plane:

If you want to describe the orientation of a crystal face or a plane of atoms within a crystal lattice, then there are series of steps that will lead you to its notation using Miller indices.

  1. The first step is to determine the fractional intercepts that the plane/face makes with each crystallographic axis. In other words, how far along the unit cell lengths does the plane intersect the axis. In the figure above, the plane intercepts each axis at exact one unit length.
  2. Step two involves taking the reciprocal of the fractional intercept of each unit length for each axis. In the figure above, the values are all 1/1.
  3. Finally the fractions are cleared (i.e., make 1 as the common denominator). The cleared fractions result in 3 integer values. These integers are designated h, k and l.
  4. These integer numbers are then parenthetically enclosed (hkl) to indicate the specific crystallographic plane within the lattice. Since the unit cell repeats in space, the (hkl) notation actually represents a family of planes, all with the same orientation. In the figure above, the Miller indices for the plane are (111).

Why go through all of these reciprocal operations?

Example of the (101) plane:

This becomes immediately apparent when we consider the case of the (101). In this case, the plane intercepts the a axis at one unit length and also the c axis at one unit length. The plane however, never intersects the b axis. In other words, it can be said that the intercept to the b axis is infinity. The intercepts are then designated as 1,infinity,1. The reciprocals are then 1/1, 1/infinity, 1/1. Knowing 1/infinity = 0 then the indices become (101).

Example of the (102) plane:

Example of the (-102) plane:

Examples of the (102) and (201) planes:

Bravais-Miller Indices

The crystals that belong to the hexagonal system can be viewed as having either a 6-fold or 3-fold  rotational symmetry axis.  One approach to viewing the symmetry content of the hexagonal system is to assign a set of three identical axes (a1, a2, a3) that are perpendicular to a plane that contains the three axes. This alternate form of describing the hexagaonal system with four crystallographic axes is known as the Bravais-Miller system. The symbolic notation for this system is given as (hkil), where the first three integers refer to the three identical axes, respectively.

In this notation, h + k = - i

  hexagonal lattice

Closures for crystallographic indices

(hkl) = parenthesis designate a crystal face or a family of planes throughout a crystal lattice.

[uvw] = square brackets designate a direction in the lattice from the origin to a point. Used to collectively include all the faces of a crystals whose intersects (i.e., edges) parallel each other. These are referred to as crystallographic zones and they represent a direction in the crystal lattice.

{hkl} = "squiggly" brackets or braces designate a set of faces that are equivalent by the symmetry of the crystal. The set of face planes results in the crystal form. {100} in the isometric class includes (100), (010), (001), (-100), (0-10) and (00-1), while for the triclinic {100} only the (100) is included.

d-spacing is defined as the distance between adjacent planes. When X-rays diffract due to interference amongst a family of similar atomic planes, then each diffraction plane may be reference by it's indices dhkl