Miller indices
/ MIL-er /
To name a flat plane slicing through a crystal, you note where it crosses the three axes and then apply one clever trick. Miller indices are the standard three-number label, written (hkl), for a crystallographic plane.
The recipe: find the plane's intercepts on the a, b, and c axes in units of the cell edges; take the reciprocal of each intercept; then clear fractions to the smallest whole integers. A plane parallel to an axis never crosses it, so its intercept is infinity and the reciprocal is 0. A plane cutting all three axes at 1 is (111); one crossing only the x-axis at 1 is (100), a cube face. In a cubic crystal the spacing between adjacent planes is d = a / sqrt(h^2 + k^2 + l^2).
Why bother? The planes with the most atoms packed per unit area — the close-packed planes such as FCC's (111) — are precisely the planes a crystal slips on and the planes that reflect X-rays. So (hkl) is the shared language linking structure, deformation, and diffraction. A whole family of equivalent planes is written with curly braces, like {100} for all six cube faces.
A plane cutting the x-axis at 1 and parallel to y and z has intercepts (1, infinity, infinity); the reciprocals (1, 0, 0) give the (100) plane — a cube face. A plane cutting all three axes at 1 is (111), the close-packed plane of FCC metals.
Reciprocals of intercepts give (hkl): (100) is a cube face, (111) the close-packed plane.
Take reciprocals of the intercepts — that is the whole trick. And note (hkl) with round brackets means a plane, distinct from [uvw] directions in square brackets; do not mix the bracket types.