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Article: Miller index

Miller index

Miller Indices and Crystallography

  • Miller indices are a notation system used in crystallography.
  • They represent lattice planes in crystal lattices.
  • Negative integers are denoted with a bar.
  • Miller indices can also designate reflections in X-ray crystallography.
  • They can be thought of as corresponding to planes spaced such that reflections from adjacent planes have a phase difference of one wavelength.
  • Miller indices can be determined using intercepts with axes.
  • There are two ways to define the meaning of Miller indices: via a point in the reciprocal lattice or as the inverse intercepts along the lattice vectors.
  • The indices are proportional to the inverses of the intercepts of the plane.
  • If one of the indices is zero, it means the planes do not intersect that axis.
  • The perpendicular distance between adjacent lattice planes is related to the reciprocal lattice vector.
  • In simple cubic crystals, the Miller indices and [hkℓ] both denote normals/directions in Cartesian coordinates.
  • The spacing between adjacent (hkℓ) lattice planes is determined by the lattice constant and the indices.
  • Indices in angle brackets denote a family of equivalent directions due to symmetry operations.
  • Indices in curly brackets denote a family of equivalent plane normals.
  • For face-centered cubic and body-centered cubic lattices, the Miller indices are defined relative to the lattice vectors of the cubic supercell.
  • Miller-Bravais indices are used for hexagonal and rhombohedral lattice systems.
  • They consist of four indices that obey a constraint.
  • The redundant index in the four-index scheme makes permutation symmetries apparent.
  • The (001) plane in hexagonal lattice has a 3-fold symmetry.
  • There are ad hoc schemes for indexing hexagonal lattice vectors with four indices.
  • There are related notations like {hkℓ} and [hkℓ] that denote equivalent planes or directions.
  • The reciprocal lattice vector (hkℓ) can be expressed in terms of reciprocal lattice vectors.
  • Zone indices of the direction perpendicular to a plane can be represented in triplet form.
  • Four-index zone indices sometimes mix direct-lattice and reciprocal-lattice indices.
  • Hexagonal interplanar distances have a specific form in terms of Miller indices.

Influence of Crystallographic Planes and Directions on Crystal Properties

  • Crystallographic directions are lines connecting nodes of a crystal.
  • Crystallographic planes are planes connecting nodes of a crystal.
  • Dense planes have a higher density of nodes and influence the behavior of the crystal.
  • Optical properties of condensed matter are affected by the density of nodes on crystallographic planes.
  • Adsorption, reactivity, and surface tension are sensitive to the density of nodes on crystal surfaces.
  • Pores and crystallites tend to have straight grain boundaries following dense planes.
  • Cleavage and plastic deformation (dislocations) occur more frequently on dense planes.
  • Dislocation cores spread on dense planes, reducing friction.
  • The perturbation carried by dislocations is along a dense direction.
  • Dislocation lines and loops often follow dense directions.

Integer versus Irrational Miller Indices: Lattice Planes and Quasicrystals

  • Miller indices are usually integers and have physical significance.
  • Integer Miller indices implicitly include indices with all rational ratios.
  • Lattice planes are planes with rational-ratio Miller indices and have periodic intersections with the crystal.
  • Planes with irrational-ratio Miller indices form aperiodic patterns known as quasicrystals.
  • Quasicrystals can be defined using a plane with irrational-ratio Miller indices.

Related Concepts and Notations

  • Crystal structure
  • Crystal habit
  • Kikuchi line
  • Zone axis


  • Ashcroft, Neil W.; Mermin, N. David (1976). Solid state physics.
  • Weiss, Christian Samuel (1817). Ueber eine verbesserte Methode für die Bezeichnung der verschiedenen Flächen eines Krystallisationssystems, nebst Bemerkungen über den Zustand der Polarisierung der Seiten in den Linien der krystallinischen Structur.
  • Oxford English Dictionary Online
  • J. W. Edington (1976). Practical electron microscopy in materials science.
  • IUCr Online Dictionary of Crystallography

Miller index Data Sources

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