Miller index

A set of integers that specifies the orientation of a crystal plane or family of parallel planes relative to the crystal axes. Miller indices are derived from the plane’s axis intercepts and are written in parentheses for one plane.

Crystal lattice: A periodic arrangement of points representing the repeating structure of a crystal. Miller indices describe planes relative to the axes of this repeating framework.
Miller–Bravais indices: A four-index notation for planes and directions in hexagonal crystals. Its extra index handles hexagonal symmetry where three equivalent basal axes meet.
Bragg's law: A relation linking X-ray wavelength, crystal-plane spacing, and diffraction angle. Diffraction peaks are indexed by the lattice planes that satisfy its condition.
Crystallographic direction: A direction in a crystal lattice expressed using integer coordinates along crystallographic axes. Directions use square brackets, whereas planes use parentheses and reciprocal intercepts.
Crystallographic axis: A reference direction used to describe positions and orientations in a crystal. The chosen crystal axes determine the intercepts from which Miller indices are calculated.
Weiss zone law: A crystallographic relation connecting a plane’s Miller indices to directions lying within it. It tests whether a crystallographic direction lies in a plane with given indices.
Interplanar spacing: The perpendicular distance between adjacent parallel planes in a crystal. Plane indices help calculate spacing, though the formula depends on lattice geometry.
Miller index notation: The conventions for writing crystallographic plane and direction indices with brackets and bars. Parentheses, braces, and overbars distinguish a plane, a family, and negative indices.
Reciprocal lattice: A lattice in reciprocal space whose vectors encode periodicity and directions in a crystal. A plane’s Miller indices give the direction of its reciprocal-lattice normal.
Plane normal: A vector perpendicular to a plane. In reciprocal space, the Miller-index vector points along the plane normal.

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