5. Williamson-Hall Method

Separating Size and Strain Broadening

In addition to finite crystallite sizes, peak broadening can also be caused by microstrains in the crystal lattice. These strains represent local displacements of atoms from their ideal positions, caused by defects like vacancies, dislocations, or grain boundary stresses.

Because size and strain broadening occur simultaneously, they must be separated to prevent underestimating crystallite sizes. The Williamson-Hall method achieves this by analyzing how broadening varies with the diffraction angle.

The Williamson-Hall Relation

While crystallite size broadening varies with 1/cos(θ), microstrain broadening varies with tan(θ). Assuming these contributions add linearly, the total FWHM (β_tot) can be expressed as:

Where:

• β_tot: The measured peak FWHM in radians, corrected for instrument broadening.
• θ: The Bragg angle of the peak centroid.
• K: The crystallite shape factor (typically $0.9$).
• λ: The X-ray source wavelength.
• D: The strain-corrected mean crystallite size.
• ε: The lattice microstrain parameter.

Plotting and Interpretation

By plotting Y = β_tot cos(θ) on the y-axis against X = 4 sin(θ) on the x-axis, we can fit the data with a linear regression line (y = mx + c):

• Slope (m): The slope of the line represents the lattice microstrain ($\varepsilon$). A positive slope indicates tensile strain, while a negative slope indicates compressive strain.
• Y-Intercept (c): The y-intercept represents the size term ($K\lambda / D$). We can calculate the strain-corrected crystallite size using:
  D = Kλ / c

Application Guidelines

To obtain reliable results, you should fit at least 3 to 5 peaks from the same phase. High scatter in the data points suggests anisotropic microstrains or directional grain growth, in which case more advanced models (like modified Williamson-Hall equations) may be required.