Rietveld method¶

The Rietveld method compares a calculated powder pattern with measured intensities over the included observations. It fits a crystallographic model together with the relevant experiment parameters.

Calculated powder pattern¶

A useful decomposition is

\[ y_i^{\rm calc}=b(x_i)+\sum_p s_p\sum_{h\in p}m_h\,L_h\,P_h\,A_h\,O_h\,|F_h|^2\,\Omega_p(x_i-x_h). \]

Here \(p\) labels a linked phase, \(s_p\) its scale, \(h\) a reflection, \(m_h\) multiplicity, \(L_h\) the geometry-dependent Lorentz factor, \(P_h\) polarization where applicable, \(A_h\) absorption, \(O_h\) preferred orientation, \(F_h\) the structure factor, \(\Omega\) the selected peak profile and \(b\) the background. This is a model decomposition: factors that do not apply to the selected experiment are unity, and instrument-specific intensity-unit conventions are carried with its scale.

Follow the terms to crystallography, scattering, profiles, absorption, orientation, Lorentz/polarization and background. Single-crystal reflection calculations use the corresponding integrated-intensity model, not a powder-profile sum.

Least-squares objective¶

\[ r_i(\mathbf p)=\frac{y_i^{\rm calc}(\mathbf p)-y_i^{\rm obs}}{\sigma_i},\qquad S(\mathbf p)=\sum_{i\in\mathcal I}r_i^2. \]

\(\mathcal I\) is the included observation set and \(\sigma_i>0\) its standard uncertainty. Reversing every residual's sign leaves \(S\) unchanged. The displayed observed-minus-calculated difference curve can therefore have the opposite sign to the solver residual.

Only independent free parameters are optimisation variables. Symmetry, constraints, the selected experiment and model type determine which quantities are independent and affect the observations. See minimization.

Single and joint fitting¶

A single fit uses one experiment. A joint fit combines the residuals of several experiments, sharing their common structural quantities while retaining experiment-specific terms. If experiment weights \(v_e\) are applied, its residual block is multiplied by \(\sqrt{v_e}\) and its objective contribution by \(v_e\).

Inspect each experiment after a joint fit. Combined agreement can hide a systematic mismatch in one dataset. Reliability factors distinguish per-experiment and pooled summaries.

Sequential and independent scans¶

Both scan modes run a sequence of fits against a template experiment. Their distinction is the starting model, not parallel execution. See the fitting-mode guide.

Sequential scans¶

Each dataset starts from the preceding fitted result. This carries the evolving model forward through the series.

Independent scans¶

Every dataset starts from the same template values, so the starting model does not depend on the preceding fit.

Intensity extraction¶

Proposed. These methods are not available fitting modes in this reference snapshot. See Pattern fitting for current alternatives.

Le Bail intensity extraction¶

Reflection intensities are updated by iterative decomposition of the measured powder pattern, alongside profile fitting.

Pawley intensity extraction¶

Reflection intensities are adjustable variables in a least-squares fit of the powder pattern.

The distinction is described in the IUCr intensity-extraction overview.

Interpretation¶

Fitting improves agreement under the model and its uncertainties. Convergence does not establish that the phase model, background, calibration or uncertainty estimates are correct. Use the practical fit assessment and retain the starting project.

References¶

McCusker, L. B., Von Dreele, R. B., Cox, D. E., Louër, D. & Scardi, P. (1999). Rietveld refinement guidelines. Journal of Applied Crystallography, 32, 36–50.