Miller index
Miller indices form a notation system in crystallography for planes in crystal lattices.
In particular, a family of lattice planes is determined by three integers h, k, and ℓ, the Miller indices. They are written, and denote the family of planes orthogonal to, where are the basis of the reciprocal lattice vectors. 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.
- , 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
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 h, k, ℓ, 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.
Case of 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.
Case of hexagonal and rhombohedral structures
With hexagonal and rhombohedral lattice systems, it is possible to use the Bravais-Miller system, which uses four indices that obey the constraintHere 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 don't 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.
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 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 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.
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.