What is the difference between Scherrer equation and Williamson-Hall method?

The Scherrer equation assumes all peak broadening is due to crystallite size. The Williamson-Hall method uses multiple peaks to separate broadening into two components: size broadening (independent of angle) and strain broadening (dependent on angle).

What does the slope represent in a Williamson-Hall plot?

In a standard Williamson-Hall plot (Uniform Deformation Model), the slope of the linear regression represents the lattice micro-strain (ε). A steeper slope indicates higher internal strain in the crystal lattice.

Why do I need at least 3 peaks for Williamson-Hall analysis?

Since Williamson-Hall analysis relies on linear regression to find a slope and intercept, using only two points creates a mathematically perfect but statistically meaningless line. At least 3 peaks (preferably 5+) are required to assess the quality of the fit (linearity) and obtain reliable values.

What does a negative intercept mean in Williamson-Hall analysis?

A negative y-intercept is physically impossible as it would imply a negative crystallite size. This usually indicates over-correction of instrumental broadening, very large crystallites (>100 nm) where size broadening is negligible, or significant deviation from the isotropic strain assumption.

Williamson-Hall Plotter

Separate crystallite size and lattice strain from multi-peak XRD data

Global Parameters

Global

X-ray Wavelength (λ) Treated as exact

Wavelength must be greater than zero

Shape Factor (K) 0.62–1.0 typical

K depends on crystallite shape and peak profile assumptions.

K must be between 0.62 and 1.0 for physical validity

Peak Data

2θ (°)	β_obs (°)	β_inst (°)

Enter at least 3 peaks for meaningful regression. β values are FWHM in degrees. β_inst is optional (from standard like LaB₆). Tip: Paste data directly from Excel.

Results

0 pts

Crystallite Size (D)

— nm

Micro-Strain (ε)

—

Fit Quality (R²): —

Interpreting results: A good linear fit (R² > 0.95) supports the UDM assumption. Significant scatter suggests anisotropic strain or measurement issues.

The Williamson-Hall Method

While the Scherrer equation assumes all peak broadening is due to small crystallite size, the Williamson-Hall (W-H) method deconvolutes broadening into two components:

Size Broadening: Independent of diffraction angle (1/cos θ dependence)
Strain Broadening: Increases with diffraction angle (tan θ dependence)

The W-H Equation

The Uniform Deformation Model (UDM) combines these as:

β_tot cos θ = ε (4 sin θ) + Kλ/D

This has the form y = mx + c, where:

Y-axis (β cos θ): Total broadening contribution
X-axis (4 sin θ): Angle dependence
Slope (m = ε): Micro-strain in the lattice
Y-intercept (c = Kλ/D): Yields crystallite size D = Kλ/c

Interpreting the Plot

Positive slope: Tensile micro-strain (most common)
Negative slope: Compressive micro-strain (less common, verify data quality)
Near-zero slope: Minimal strain; Scherrer equation may suffice
Poor linearity (low R²): Anisotropic strain, mixed phases, or peak fitting issues

Negative intercept warning: A negative y-intercept is physically impossible (implies negative size). This typically indicates over-correction of instrumental broadening, crystallites >100 nm where size broadening is negligible, or violation of the isotropic strain assumption.

Data Quality Guidelines

Minimum peaks: At least 3 reflections; 5+ recommended for statistical reliability
Angular range: Include peaks across a wide 2θ range for better slope determination
Peak selection: Use well-resolved, symmetric peaks; avoid overlapping or asymmetric reflections
Instrumental correction: Measure β_inst from a strain-free standard (LaB₆, Si, corundum) at similar angles

Limitations

Isotropic assumption: UDM assumes uniform strain in all crystallographic directions.
Gaussian profile assumption: The instrumental correction uses Gaussian subtraction.
Unweighted regression: This tool performs unweighted linear regression.
Size range: Most reliable for crystallites 10–100 nm.

Sources & Citations

Williamson, G. K., & Hall, W. H. (1953). X-ray line broadening from filed aluminium and wolfram. Acta Metallurgica, 1(1), 22–31. doi:10.1016/0001-6160(53)90006-6

Mote, V. D., Purushotham, Y., & Dole, B. N. (2012). Williamson-Hall analysis in estimation of lattice strain in nanometer-sized ZnO particles. Journal of Theoretical and Applied Physics, 6, 6.

Zak, A. K., et al. (2011). X-ray analysis of ZnO nanoparticles by Williamson-Hall and size-strain plot methods. Solid State Sciences, 13(1), 251–256.

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