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# Miller index

Miller indices are a notation system in crystallography for planes and directions in crystal lattices.

In particular, a family of lattice planes is determined by three integers $\ell$, $m\,$, and $n\,$, the Miller indices. They are written $(\ell m n)$ and denote planes orthogonal to a direction $(\ell, m, n)$ in the basis of the reciprocal lattice vectors. By convention, negative integers are written with a bar, as in $\bar{3}$ for − 3. The integers are usually written in lowest terms, i.e. their greatest common divisor should be 1.

There are also several related notations. $[\ell m n]$, with square instead of round brackets, denotes a direction in the basis of the direct lattice vectors instead of the reciprocal lattice. The notation $\{\ell m n\}$ denotes all planes that are equivalent to $(\ell m n)$ by the symmetry of the crystal. Similarly, the notation $\langle \ell m n \rangle$ denotes all directions that are equivalent to $[\ell m n]$ 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.

## 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 $\mathbf{a}_1$, $\mathbf{a}_2$, and $\mathbf{a}_3$ as described above. Given these, the three primitive reciprocal lattice vectors are also determined (denoted $\mathbf{b}_1$, $\mathbf{b}_2$, and $\mathbf{b}_3$).

Then, given the three Miller indices $\ell,m,n$, $(\ell m n)$ denotes planes orthogonal to: $\ell \mathbf{b}_1 + m \mathbf{b}_2 + n \mathbf{b}_3 .$

That is, $(\ell m n)$ 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, $(\ell m n)$ denotes a plane that intercepts the three points $\mathbf{a}_1 / \ell$, $\mathbf{a}_2 / m$, and $\mathbf{a}_3 / n$, 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 $[\ell m n]$ denotes the direction: $\ell \mathbf{a}_1 + m \mathbf{a}_2 + n \mathbf{a}_3 .$

That is, it uses the direct lattice basis instead of the reciprocal lattice. Note that $[\ell m n]$ is not generally normal to the $(\ell m n)$ planes, except in a cubic lattice as described below.

## Case of the cubic structures

For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length; similarly for the reciprocal lattice. So, in this common case, the Miller indices $(\ell m n)$ and $[\ell m n]$ both simply denote normals/directions in Cartesian coordinates.

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 Bravais lattice, and hence are again simply the Cartesian directions.

## Case of the hexagonal and rhombohedral structures

With hexagonal and rhombohedral crystal systems, it is possible to use the Bravais-Miller index which has 4 numbers (h k i l)

i = −hk

where h, k and l are identical to the Miller index.

The (001) plane has a 3-fold symmetry: it remains unchanged by a rotation of 1/3 (2π/3 rad, 120°). The ,  and the $[\bar{1}\bar{1}0]$ directions are really similar. If S is the intercept of the plane with the $[1\bar{1}0]$ axis, then

i = 1/S

i is redundant and not necessary.

## 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 are 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 is important to determine the planes and thus to have a notation system.