indexing

A powder pattern hands you a row of peaks but no labels: each peak comes from some family of planes, but which one? Indexing is the detective step that assigns the correct three integers (hkl) to every peak and, from that assignment, works out the unit cell. Get the labels right and the geometry of the crystal falls out; get them wrong and everything built on top collapses.

The clean case is cubic. There the plane-spacing rule 1/d^2 = (h^2 + k^2 + l^2)/a^2 combines with Bragg's law to give sin^2(theta) proportional to the integer (h^2 + k^2 + l^2). So the ratios of sin^2(theta) for the peaks are ratios of whole numbers, and you can read off the allowed sums 1, 2, 3, 4, 5, 6, 8, 9, ... — notice 7 is missing, because no three integers square-sum to 7. The exact set of allowed sums also fingerprints the lattice type: a simple cubic, body-centred, and face-centred cell each permit a different subset (for example FCC allows only sums where h, k, l are all even or all odd). Lower-symmetry cells (tetragonal down to triclinic) are much harder and are handled by auto-indexing programs such as ITO, DICVOL, and TREOR.

Indexing yields the crystal system and the lattice parameters, and it is the prerequisite for Rietveld refinement and for any attempt at structure solution from powder. Be honest that it is harder than it looks: for low-symmetry cells, or when an impurity phase sneaks a few extra peaks into the pattern, a wrong cell can fit part of the data and send you down a blind alley. That is why indexing programs report a figure of merit to flag suspicious or accidental solutions.