7. Rietveld Refinement Guide

What is Rietveld Refinement?

Rietveld refinement, formulated by Hugo Rietveld in 1969, is a technique that fits a complete theoretical structure model to an entire measured powder diffraction pattern. Instead of fitting individual peaks independently, the method refines crystallographic and instrument parameters simultaneously using a least-squares approach.

The Refinement Equation

The algorithm minimizes the difference between the observed intensity (y_i) and the calculated intensity (y_ci) at every step (i) of the diffraction scan:

Where $w_i$ is a statistical weighting factor ($w_i = 1 / y_i$). The calculated intensity is modeled as:

In this equation, s is the scale factor, S_p represents phase contributions, L_k combines the Lorentz-Polarization and multiplicity factors, F_k is the structure factor, φ is the peak profile function, P_k is the preferred orientation correction, and y_bi is the background intensity.

Key Refined Parameters

During a refinement session, the user adjusts parameters in a specific sequence to prevent the least-squares solver from diverging:

• 1. Scale Factor & Background: Matches the overall intensity level and adjusts the baseline curve.
• 2. Zero-Shift & Lattice Parameters: Aligns the calculated peak positions with the experimental reflections.
• 3. Profile Parameters (U, V, W): Adjusts peak widths and shapes using Caglioti equations.
• 4. Atomic Coordinates & Site Occupancies: Modifies peak intensities by refining atom positions within the unit cell.

Evaluating Fit Quality

The quality of the refinement is evaluated using several R-factors:

• Rwp (Weighted Profile R-factor): The most statistically meaningful indicator, representing the residual minimized by the solver.
• Rexp (Expected R-factor): The theoretically best possible fit based on data statistics.
• GoF (Goodness of Fit, $\chi^2$): Calculated as $(Rwp / Rexp)^2$. A GoF close to 1.0 indicates a statistically perfect fit.