Planes and Directions

Miller Indices for Crystal Planes

1. The plane intercepts at 3a₁, 2a₂, 2a₃
2. Reciprocals: 1/3, 1/2, 1/2
3. The smallest three integers having the same ratio: 2,3,3
4. The indices of the plane is (233)—(hkl) plane notation.

Common errors:
1) Forget to invert the intercepts
2) Forget to enclose in curvilinear brackets.

Indexing Rules:

• For intercept at infinity, the corresponding index is zero.
• If a plane cuts an axis on the negative side of the origin, the corresponding index is negative, indicated by placing a minus sign above the index:

• If the plane passes through the origin of coordinates, the origin of coordinates must be moved to a lattice point not on the plane to be indexed.
• All planes which fold into each other upon application of crystalsymmetry operations* cannot be distinguished from each other by any physical measurement and therefore are said to be “equivalent”.
  A group of equivalent planes is denoted by braces around the indices.
• For example

• are equivalent planes and are collectively denoted as planes.
  {100} planes.

Example: Planes

Distance between adjacent (hkl) planes

In cubic crystal structures, the interplanar spacing between two closest parallel planes with the same Miller indices is designated d_hkl

Example 1:

Example 2:

The plane intercepts at 2a₁, 2a₂, 2a₃

• Reciprocals:
  1/2, 1/2, 1/2, the indices of the plane are (111)