Miller index

Miller indices form a notation system in crystallography for lattice planes in crystal lattices.
In particular, a family of lattice planes of a given Bravais lattice is determined by three integers h, k, and ℓ, the Miller indices. They are written, and denote the family of lattice planes orthogonal to, where are the basis or primitive translation vectors of the reciprocal lattice for the given Bravais lattice. This is based on the fact that a reciprocal lattice vector is the wavevector of a plane wave in the Fourier series of a spatial function which periodicity follows the original Bravais lattice, so wavefronts of the plane wave are coincident with parallel lattice planes of the original lattice. Since a measured scattering vector in X-ray crystallography, with as the outgoing X-ray wavevector and as the incoming X-ray wavevector, is equal to a reciprocal lattice vector as stated by the Laue equations, the measured scattered X-ray peak at each measured scattering vector is marked by Miller indices.
By convention, negative integers are written with a bar, as in for −3. The integers are usually written in lowest terms, i.e. their greatest common divisor should be 1. Miller indices are also used to designate reflections in X-ray crystallography. In this case the integers are not necessarily in lowest terms, and can be thought of as corresponding to planes spaced such that the reflections from adjacent planes would have a phase difference of exactly one wavelength, regardless of whether there are atoms on all these planes or not.
There are also several related notations:

the notation denotes the set of all planes that are equivalent to by the symmetry of the lattice.

In the context of crystal directions, the corresponding notations are:

with square instead of round brackets, denotes a direction in the basis of the direct lattice vectors instead of the reciprocal lattice; and
similarly, the notation denotes the set of all directions that are equivalent to by symmetry.

Note, for Laue–Bragg interferences

lacks any bracketing when designating a reflection

Miller indices were introduced in 1839 by the British mineralogist William Hallowes Miller, although an almost identical system had already been used by German mineralogist Christian Samuel Weiss since 1817. The method was also historically known as the Millerian system, and the indices as Millerian, although this is now rare.
The Miller indices are defined with respect to any choice of unit cell and not only with respect to primitive basis vectors, as is sometimes stated.

Definition

There are two equivalent ways to define the meaning of the Miller indices: via a point in the reciprocal lattice, or as the inverse intercepts along the lattice vectors. Both definitions are given below. In either case, one needs to choose the three lattice vectors a₁, a₂, and a₃ that define the unit cell. Given these, the three primitive reciprocal lattice vectors are also determined.
Then, given the three Miller indices denotes planes orthogonal to the reciprocal lattice vector:
That is, simply indicates a normal to the planes in the basis of the primitive reciprocal lattice vectors. Because the coordinates are integers, this normal is itself always a reciprocal lattice vector. The requirement of lowest terms means that it is the shortest reciprocal lattice vector in the given direction.
Equivalently, denotes a plane that intercepts the three points a₁/h, a₂/k, and a₃/ℓ, or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane, in the basis of the lattice vectors. If one of the indices is zero, it means that the planes do not intersect that axis.
Considering only planes intersecting one or more lattice points, the perpendicular distance d between adjacent lattice planes is related to the reciprocal lattice vector orthogonal to the planes by the formula:.
The related notation denotes the direction:
That is, it uses the direct lattice basis instead of the reciprocal lattice. Note that is not generally normal to the planes, except in a cubic lattice as described below.

Cubic structures

For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length, as are those of the reciprocal lattice. Thus, in this common case, the Miller indices and both simply denote normals/directions in Cartesian coordinates.
For cubic crystals with lattice constant a, the spacing d between adjacent lattice planes is
Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes:

Indices in angle brackets such as ⟨100⟩ denote a family of directions which are equivalent due to symmetry operations, such as , , or the negative of any of those directions.
Indices in curly brackets or braces such as denote a family of plane normals which are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.

For face-centered cubic and body-centered cubic lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions.

Hexagonal and rhombohedral structures

For hexagonal and rhombohedral lattice systems, the Bravais–Miller system is typically used, which uses four indices that obey the constraint
Here h, k and ℓ are identical to the corresponding Miller indices, and i is a redundant index.
This four-index scheme for labeling planes in a hexagonal lattice makes permutation symmetries apparent. For example, the similarity between ≡ and ≡ is more obvious when the redundant index is shown.
In the figure at right, the plane has a 3-fold symmetry: it remains unchanged by a rotation of 1/3. The , and the directions are really similar. If S is the intercept of the plane with the axis, then
There are also ad hoc schemes for indexing hexagonal lattice vectors with four indices. However they do not operate by similarly adding a redundant index to the regular three-index set.
For example, the reciprocal lattice vector as suggested above can be written in terms of reciprocal lattice vectors as. For hexagonal crystals this may be expressed in terms of direct-lattice basis-vectors a₁, a₂ and a₃ as
Hence zone indices of the direction perpendicular to plane are, in suitably normalized triplet form, simply. When four indices are used for the zone normal to plane, however, the literature often uses instead. 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 on the left.
And, note that for hexagonal interplanar distances, they take the form

Crystallographic planes and directions

Crystallographic directions are lines linking nodes of a crystal. Similarly, crystallographic planes are planes linking nodes. Some directions and planes have a higher density of nodes; these dense planes have an influence on the behavior of the crystal:

optical properties: in condensed matter, 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: adsorption and chemical reactions can occur at atoms or molecules on crystal surfaces, 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
* Pores and crystallites tend to have straight grain boundaries following dense planes
*cleavage
dislocations
*the dislocation core tends to spread on dense planes ; this reduces the friction, the sliding occurs more frequently on dense planes;
*the perturbation carried by the dislocation 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.

Non-integer Miller indices

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 are not necessarily integers.

Lattice planes

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 have rational ratios are of special interest is that these are the lattice planes: they are the only planes whose intersections with the crystal are 2d-periodic.

Quasicrystals

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. This construction corresponds precisely to the standard "cut-and-project" method of defining a quasicrystal, using a plane with irrational-ratio Miller indices.