Miller index
(Note that the plane is not always orthogonal to the linear combination of direct or original lattice vectors
as the outgoing (scattered from a crystal lattice) X-ray wavevector and
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.
There are also several related notations:[1] In the context of crystal directions (not planes), the corresponding notations are: Note, for Laue–Bragg interferences Miller indices were introduced in 1839 by the British mineralogist William Hallowes Miller, although an almost identical system (Weiss parameters) had already been used by German mineralogist Christian Samuel Weiss since 1817.
Given these, the three primitive reciprocal lattice vectors are also determined (denoted b1, b2, and b3).
denotes planes orthogonal to the reciprocal lattice vector: That is, (hkℓ) 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, (hkℓ) 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 (the intercept is "at infinity").
For cubic crystals with lattice constant a, the spacing d between adjacent (hkℓ) lattice planes is (from above) 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: 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.
With hexagonal and rhombohedral lattice systems, it is possible to use the Bravais–Miller system, which uses four indices (h k i ℓ) 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 (110) ≡ (1120) and (120) ≡ (1210) 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°).
If S is the intercept of the plane with the [110] 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 do not operate by similarly adding a redundant index to the regular three-index set.
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 (hkℓ) are, in suitably normalized triplet form, simply
When four indices are used for the zone normal to plane (hkℓ), however, the literature often uses
[4] 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.
And, note that for hexagonal interplanar distances, they take the form Crystallographic directions are lines linking nodes (atoms, ions or molecules) of a crystal.
Ordinarily, Miller indices are always integers by definition, and this constraint is physically significant.
To understand this, suppose that we allow a plane (abc) 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 (hkℓ) by scaling a, b and c appropriately: divide by the largest of the three numbers, and then multiply by the least common denominator.
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.)
