Editing Miller index (section)

==Definition==
[[Image:Indices miller plan definition.svg|thumb|300px|Examples of determining indices for a plane using intercepts with axes; left (111), right (221)]]
There are two equivalent ways to define the meaning of the Miller indices:<ref name="Ash"/> 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 '''a'''<sub>1</sub>, '''a'''<sub>2</sub>, and '''a'''<sub>3</sub> that define the unit cell (note that the conventional unit cell may be larger than the primitive cell of the [[Bravais lattice]], as the [[Miller_index#Case_of_cubic_structures|examples below]] illustrate). Given these, the three primitive reciprocal lattice vectors are also determined (denoted '''b'''<sub>1</sub>, '''b'''<sub>2</sub>, and '''b'''<sub>3</sub>).

Then, given the three Miller indices <math> h, k, \ell, (hk\ell) </math> denotes planes orthogonal to the reciprocal lattice vector:
:<math> \mathbf{g}_{hk\ell} = h \mathbf{b}_1 + k \mathbf{b}_2 + \ell \mathbf{b}_3 .</math>
That is, (''hkℓ'') simply indicates a normal to the planes in the [[Basis (linear algebra)|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 '''a'''<sub>1</sub>/''h'', '''a'''<sub>2</sub>/''k'', and '''a'''<sub>3</sub>/''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").

Considering only (''hkℓ'') planes intersecting one or more lattice points (the ''lattice planes''), the perpendicular distance ''d'' between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula: <math>d = 2\pi / |\mathbf{g}_{h k \ell}|</math>.<ref name="Ash"/>

The related notation [hkℓ] denotes the ''direction'':
:<math>h \mathbf{a}_1 + k \mathbf{a}_2 + \ell \mathbf{a}_3 .</math>
That is, it uses the direct lattice basis instead of the reciprocal lattice. Note that [hkℓ] is ''not'' generally normal to the (''hkℓ'') planes, except in a cubic lattice as described below.