WilliamsonHall

The Williamson–Hall method, commonly referred to as Williamson–Hall analysis, is a widely used approach in X-ray diffraction to separate size broadening and strain broadening of diffraction peaks. It was developed by Williamson and Hall in 1953 and remains a standard tool for estimating crystallite size and lattice microstrain in polycrystalline materials.

The method starts from the observation that peak broadening beta (in radians) arises from both finite crystallite size and lattice strain. For a particular reflection at angle theta with X-ray wavelength lambda and shape factor K (≈0.9), the broadening is approximated by beta = K lambda / (D cos theta) + 4 epsilon tan theta, where D is the average crystallite size and epsilon is the microstrain. After correcting beta for instrumental broadening, this equation is rearranged to beta cos theta = K lambda / D + 4 epsilon sin theta.

A Williamson–Hall plot is then made by graphing beta cos theta (vertical) against sin theta (horizontal) for multiple reflections. The data should lie on a straight line if strain and size broadening are isotropic. The intercept on the beta cos theta axis equals K lambda / D, allowing D to be obtained as D = K lambda / intercept, while the slope equals 4 epsilon, giving the microstrain as epsilon = slope / 4.

Applications include characterizing nanocrystalline metals and ceramics, and evaluating processing effects on crystallite size and internal strain. Limitations include the assumption of isotropic, uniform size and strain distributions and the need for accurate instrumental broadening correction. Anisotropy, preferred orientation, and complex defect structures can bias results, in which case more advanced methods (e.g., Warren–Averbach, size–strain plots) may be used.