Miller index

Miller indices form a notation system in crystallography for planes in crystal (Bravais) lattices.

In particular, a family of lattice planes is determined by three integers h, k, and ℓ, the Miller indices. They are written (hkℓ), and denote the family of planes orthogonal to ${\displaystyle h\mathbf {b_{1}} +k\mathbf {b_{2}} +\ell \mathbf {b_{3}} }$, where ${\displaystyle \mathbf {b_{i}} }$ are the basis of the reciprocal lattice vectors. (Note that the plane is not always orthogonal to the linear combination of direct lattice vectors ${\displaystyle h\mathbf {a_{1}} +k\mathbf {a_{2}} +\ell \mathbf {a_{3}} }$ because the reciprocal lattice vectors need not be mutually orthogonal.) By convention, negative integers are written with a bar, as in 3 for −3. The integers are usually written in lowest terms, i.e. their greatest common divisor should be 1.

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