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 for . The integers are usually written in lowest terms, i.e. their greatest common divisor should be 1.
There are also several related notations. , with square instead of round brackets, denotes a direction in the basis of the direct lattice vectors instead of the reciprocal lattice. The notation 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 are not primitive (e.g., as in body-centered and face-centered crystals).
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 , , and as described above. Given these, the three primitive reciprocal lattice vectors are also determined (denoted , , and ).
Then, given the three Miller indices , denotes planes orthogonal to:
Equivalently, denotes a plane that intercepts the three points , , and , 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 (the intercept is "at infinity").
The related notation denotes the direction:
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.
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 (001) plane has a 3-fold symmetry: it remains unchanged by a rotation of 1/3 (2π/3 rad, 120°). 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 (e.g. in the transmission electron microscopy literature) for indexing hexagonal lattice vectors (rather than reciprocal lattice vectors or planes) 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 (hkl) as suggested above can be written as ha*+kb*+lc* if the reciprocal-lattice basis-vectors are a*, b*, and c*. For hexagonal crystals this may be expressed in terms of direct-lattice basis-vectors a, b and c as
Hence zone indices of the direction perpendicular to plane (hkl) are, in suitably-normalized triplet form, simply [2h+k,h+2k,l(3/2)(a/c)2]. When four indices are used for the zone normal to plane (hkl), however, the literature often uses [h,k,-h-k,l(3/2)(a/c)2] 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 (normally in round or curly brackets) on the left.
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:
For all these reasons, it is important to determine the planes and thus to have a notation system.
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 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. 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.)