Everything about Miller Index totally explained
Miller indices are a notation system in
crystallography for planes and directions in
crystal (Bravais) lattices.
In particular, a family of
lattice planes is determined by three
integers
,
, and
, the
Miller indices. They are written
and denote planes orthogonal to a direction
in the
basis of the
reciprocal lattice vectors. By convention,
negative integers are written with a bar, as in
denotes all planes that are equivalent to
by the symmetry of the crystal. Similarly, the notation
denotes all directions that are equivalent to
by symmetry.
Miller indices were introduced in 1839 by the British mineralogist
William Hallowes Miller. The method was also historically known as the Millerian system, and the indices as Millerian, although this is now rare.
The precise meaning of this notation depends upon a choice of lattice vectors for the crystal, as described below. Usually, three primitive lattice vectors are used. However, for cubic crystal systems, the cubic lattice vectors are used even when they're not primitive (for example, as in body-centered and face-centered crystals).
Definition
There are two equivalent ways to define the meaning of the Miller indices:. Thus as you can see, four-index zone indices in square or angle brackets sometimes mix a single direct-lattice index on the right with reciprocal-lattice indices (normally in round or curly brackets) on the left.
The crystallographic planes and directions
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're surrounded by other similar species; the surface tension of an interface thus varies according to the density on the surface
- 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's important to determine the planes and thus to have a notation system.
Integer vs. irrational Miller indices: Lattice planes and quasicrystals
Ordinarily, Miller indices are always integers by definition, and this constraint is physically significant. To understand this, suppose that we allow a plane
where the Miller "indices"
a,
b, and
c (defined as above) are not necessarily integers.
If
a,
b, and
c have
rational ratios, then the same family of planes can be written in terms of integer indices
by scaling
a,
b, and
c appropriately: divide by the largest of the three numbers, and then multiply by the
least common denominator. Thus, integer Miller indices implicitly include indices with all rational ratios. The reason why planes where the components (in the reciprocal-lattice basis) have rational ratios are of special interest is that these are the
lattice planes: they're the only planes whose intersections with the crystal are 2d-periodic.
For a plane
where
a,
b, and
c have
irrational ratios, on the other hand, the intersection of the plane with the crystal is
not periodic. It forms an aperiodic pattern known as a
quasicrystal. In fact, this construction corresponds precisely to the standard "cut-and-project" method of defining a quasicrystal, using a plane with irrational-ratio Miller indices. (Although many quasicrystals, such as the
Penrose tiling, are formed by "cuts" of periodic lattices in more than three dimensions, involving the intersection of more than one such
hyperplane.)
Further Information
Get more info on 'Miller Index'.
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