Miller index

Examples of directions

Miller indices are used to describe planes and directions in a crystal.

It is necessary to define a basis for the single crystal; see Primitive cell.

In case of directions, a Miller index is of the form of

[u v w]

where the integers in the square brackets represent the coordinates of the vector in the real space.

Definition of the index for a plane

In the case of planes, a Miller index is of the form

(h k l)

where the integers h, k, and l represent the x-, y-, and z-intercepts of the plane respectively:

if P, Q and R are the coordinates of the intercept of the plane (the closest to the origin) with the axes, then

h = 1/P

k = 1/Q

l = 1/R

In Miller indices, a bar above one of the numbers indicates a negative value.

The crystallographic planes and directions

Dense crystallographic planes

The crystallographic directions are fictitious lines linking nodes (atoms, ions or molecules) of a crystal. The crystallographic planes are fictitious planes linking nodes. Some directions and planes have a higher density of nodes; these dense planes have an influence on the behaviour of the crystal:

• optical properties: in condensed matter, the light "jumps" from one atom to the other with the Rayleigh scattering; the velocity of light thus varies according to the directions, whether the atoms are close or far; this gives the birefringence
• adsorption and reactivity: the adsorption and the chemical reactions occur on atoms or molecules, these phenomena are thus sensitive to the density of nodes;
• surface tension: the condensation of a material means that the atoms, ions or molecules are more stable if they are surrounded by other similar species; the surface tension of an interface thus varies according to the density on the surface
  • the pores and crystallites tend to have straight grain boundaries following dense planes
  • cleavage
• dislocations (plastic deformation)
  • the dislocation core tends to spread on dense planes (the elastic perturbation is "diluted"); this reduces the friction (Peierls-Nabarro force), the sliding occurs more frequently on dense planes;
  • the perturbation carried by the dislocation (Burgers vector) is along a dense direction: the shift of one node in a dense direction is a lesser distortion;
  • the dislocation line tends to follow a dense direction, the dislocation line is often a straight line, a dislocation loop is often a polygon.

For all these reasons, it is important to determine the planes and thus to have a notation system.

Case of the cubic structures

In case of the cubic structures, the Miller index of a plane, in parentheses such as (100), are also the coordinates of the direction of a plane normal. It stands for a vector perpendicular to the family of planes, with a length of d^-1, where d is the inter-plane spacing.

Due to the symmetries of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes:

• Coordinates in angle brackets or chevrons such as <100> denote a family of directions which are equivalent due to symmetry operations. If it refers to a cubic system, this example could mean [100], [010], [001] or the negative of any of those directions.
• Coordinates in curly brackets or braces such as {100} denote a family of plane normals which are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.

Case of the hexagonal and rhombohedral structures

Miller-Bravais index

With hexagonal and rhombohedral crystal systems, it is possible to use the Bravais-Miller index which has 4 numbers (h k i l)

i = -h-k

where h, k and l are identical to the Miller index.

The (100) plane has a 3-fold symmetry, it remains unchanged by a rotation of 1/3 (2π/3 rad, 120°). The [100], [010] and the $[{\bar {1}}{\bar {1}}0]$ directions are really similar. If S is the intercept of the plane with the $[1{\bar {1}}0]$ axis, then

i = 1/S

i is redundant and not necessary.

See also

• Crystal structure
• Crystal

External links

• http://www.ece.byu.edu/cleanroom/EW_orientation.phtml - Miller index description with diagrams