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Miller Index




In Geometry , a Miller index is 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
: ''v'' ''w''
where the Integers in the square Bracket s represent the coordinates of the Vector in the real space.

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''- Intercept s 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''


THE CRYSTALLOGRAPHIC PLANES AND DIRECTIONS


The crystallographic directions are fictitious Line s linking nodes ( Atom s, Ion s or Molecule s) of a crystal. The crystallographic Plane s 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:
  • "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 Pore s and Crystallite s tend to have straight grain boundaries following dense planes

  • --- Cleavage

  • Dislocation s ( 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 a cubic structure, 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 [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 RHOMBOHEDRAL AND STRUCTURES


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, 30°). The [010 and the directions are 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




EXTERNAL LINKS


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





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