page_title: Miller index
hierarchical_section_title: Miller index
context_page_description: Miller indices form a notation system in crystallography for planes in crystal lattices.
  In particular, a family of lattice planes is determined by three integers h, k, and ℓ, the Miller indices. They are written, and denote the family of planes orthogonal to, where are the basis of the reciprocal lattice vectors. 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. Miller indices are also used to designate reflections in X-ray crystallography. In this case the integers are not necessarily in lowest terms, and can be thought of as corresponding to planes spaced such that the reflections from adjacent planes would have a phase difference of exactly one wavelength, regardless of whether there are atoms on all these planes or not.
  There are also several related notations:
  the notation {hkℓ} denotes the set of all planes that are equivalent to by the symmetry of the lattice.
  In the context of crystal directions, the corresponding notations are:
context_section_description: 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 , where  are the basis of the reciprocal lattice vectors (note that the plane is not always orthogonal to the linear combination of direct lattice vectors  because the  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. Miller indices are also used to designate reflections in X-ray crystallography. In this case the integers are not necessarily in lowest terms, and can be thought of as corresponding to planes spaced such that the reflections from adjacent planes would have a phase difference of exactly one wavelength (2π), regardless of whether there are atoms on all these planes or not.
  There are also several related notations:
  the notation {hkℓ} denotes the set of all planes that are equivalent to (hkℓ) by the symmetry of the lattice.
  In the context of crystal directions (not planes), the corresponding notations are:
  [hkℓ], with square instead of round brackets, denotes a direction in the basis of the direct lattice vectors instead of the reciprocal lattice; and
  similarly, the notation <hkℓ> denotes the set of all directions that are equivalent to [hkℓ] by symmetry.
  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. The method was also historically known as the Millerian system, and the indices as Millerian, although this is now rare.
  The Miller indices are defined with respect to any choice of unit cell and not only with respect to primitive basis vectors, as is sometimes stated.
caption: Examples of directions
exif: {}