Editing Crystal structure (section)

=== Miller indices ===
[[File:Miller Indices Cubes.svg|class=skin-invert-image|thumb|upright=1.2|Planes with different Miller indices in cubic crystals]]
Vectors and planes in a crystal lattice are described by the three-value [[Miller index]] notation. This syntax uses the indices ''h'', ''k'', and ''ℓ'' as directional parameters.<ref name="Physics 1991">Encyclopedia of Physics (2nd Edition), [[Rita G. Lerner|R.G. Lerner]], G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3</ref>

By definition, the syntax (''hkℓ'') denotes a plane that intercepts the three points ''a''<sub>1</sub>/''h'', ''a''<sub>2</sub>/''k'', and ''a''<sub>3</sub>/''ℓ'', or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell (in the basis of the lattice vectors). If one or more of the indices is zero, the planes do not intersect that axis (i.e., the intercept is "at infinity"). A plane containing a coordinate axis is translated to no longer contain that axis before its Miller indices are determined. The Miller indices for a plane are [[integer]]s with no common factors. Negative indices are indicated with horizontal bars, as in (1{{overbar|2}}3). In an orthogonal coordinate system for a cubic cell, the Miller indices of a plane are the Cartesian components of a vector normal to the plane.

Considering only (''hkℓ'') planes intersecting one or more lattice points (the ''lattice planes''), the distance ''d'' between adjacent lattice planes is related to the (shortest) [[reciprocal lattice]] vector orthogonal to the planes by the formula
:<math>d = \frac{2\pi}  {|\mathbf{g}_{h k \ell}|}</math>