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

References

• 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