Miller indices

Miller indices are integer labels for crystallographic planes and directions, defined relative to a crystal’s axes. Plane indices encode intercepts through their reciprocals; direction indices encode vector components.

Crystal lattice: A periodic arrangement of points representing the translational symmetry of a crystal. Miller indices describe orientations within this periodic framework.
Plane intercepts: The points where a plane crosses the coordinate axes of a chosen basis. Taking reciprocal intercepts produces the indices for a crystallographic plane.
Bragg's law: The condition relating X-ray wavelength, scattering angle, and spacing between crystal planes for constructive interference. Diffraction peaks are assigned to indexed plane families through their spacings.
Miller–Bravais indices: A four-index notation for directions and planes in hexagonal crystals. It adds a redundant basal-plane axis to make hexagonal symmetry explicit.
Cubic crystal system: A crystal system with three equal-length, mutually perpendicular lattice axes. Its symmetry makes indexed plane and direction geometry easy to visualize.
Unit cell: A repeating volume whose translations can generate an entire crystal lattice. Its crystallographic axes provide the reference for plane and direction indices.
Reciprocal space: A mathematical space in which spatial periodicity is represented by reciprocal-length vectors. Plane indices act as coordinates for normals in this space.
X-ray crystallography: A method for determining crystal structures from the diffraction of X-rays by periodic matter. Miller indices label the plane families responsible for diffraction reflections.
Zone axis: A crystallographic direction parallel to the intersection of a set of lattice planes. Zone notation relates planes sharing a direction rather than labeling one plane alone.
(100) plane: A cubic lattice plane that intercepts one axis at one lattice parameter and is parallel to the other two. It illustrates a plane with one nonzero Miller index.

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