Rietveld refinement
/ REET-velt /
Older ways of analysing a powder pattern picked out one peak at a time and measured it. Rietveld refinement does something bolder: it builds a complete calculated pattern from a structural model of the crystal and then adjusts the model until the calculated curve lies on top of the measured curve everywhere at once, point by point across the whole diffractogram. It is a least-squares fit of the entire pattern rather than of a handful of peaks.
The model has two kinds of adjustable parameters. Structural ones describe the crystal itself: the lattice parameters, the fractional coordinates of each atom, site occupancies, and atomic displacement (thermal) factors. Profile ones describe how the pattern is drawn: peak shape and width as they vary with angle, the background, an overall scale factor, and corrections such as preferred orientation. The software computes the intensity at every 2-theta step and minimises the weighted sum of (observed minus calculated) squared. You judge success by residual R-factors and a goodness-of-fit that ideally settles near one, together with a difference curve that should be flat.
The payoff is that you can extract accurate lattice parameters, atomic positions, and quantitative phase fractions (how much of each phase in a mixture) straight from powder data, without ever growing a single crystal. Be clear-eyed about the crucial caveat: Rietveld refines a model you already supply, so you must start from a roughly correct structure. It cannot invent a structure from nothing. Worse, a wrong model can sometimes be coaxed to a deceptively good R-factor — a false minimum — so a low R-factor alone is never proof that the structure is right.
To measure how much rutile and anatase (both TiO2) sit in a paint pigment, you Rietveld-refine both structures against the powder pattern at once; the refined scale factors give the weight fraction of each phase, and the difference curve flattens as the fit converges.
Fit the whole pattern, not single peaks — that is what makes powder data quantitative.
Rietveld refines, it does not solve — you must feed it a nearly correct starting model. And a good R-factor can hide a wrong structure (a false minimum), so always inspect the difference curve and chemical sense.