Crystallite size and strain calculation from Williamson hall plot from XRD Data

Crystallite size and strain are two important parameters that can be calculated from the Williamson-Hall (WH) plot obtained from X-ray diffraction (XRD) data. The WH plot is a graphical representation of the XRD peak broadening due to the crystallite size and strain in the sample. The slope of the WH plot provides the crystallite size, while the y-intercept provides the strain.

The WH plot is obtained by plotting the peak width (β) as a function of the sinθ, where θ is the diffraction angle and β is the full width at half maximum (FWHM) of the diffraction peak. The WH plot is given by the equation:

β cosθ = Kλ/D + 4ε sinθ

where K is the Scherrer constant, λ is the X-ray wavelength, D is the crystallite size, and ε is the strain. The Scherrer constant is a constant that depends on the shape of the crystallite and the X-ray wavelength used.

To calculate the crystallite size and strain from the WH plot, we first need to obtain the peak width (β) and sinθ values for the diffraction peak. This can be done using XRD analysis software, such as MATCH or X'Pert HighScore. The software can fit the diffraction peak and provide the FWHM and diffraction angle values.

Next, the β cosθ values can be calculated by multiplying the FWHM by the cosθ value. The β cosθ values are then plotted against the sinθ values to obtain the WH plot. The slope of the WH plot provides the crystallite size (D), while the y-intercept provides the strain (ε).

The crystallite size (D) can be calculated using the following equation:

D = Kλ/(β cosθ)

where K is the Scherrer constant, λ is the X-ray wavelength, β is the FWHM, and θ is the diffraction angle. The Scherrer constant depends on the shape of the crystallite and the X-ray wavelength used. The calculated crystallite size provides an average size of the crystallites in the sample.

The strain (ε) can be calculated using the y-intercept of the WH plot, which is given by:

4ε = β cosθ/sinθ

The strain provides information about the distortion or deformation of the crystal lattice due to external factors, such as temperature, pressure, or defects in the crystal structure.

It is important to note that the WH plot assumes that the broadening of the XRD peak is due to only crystallite size and strain. However, there may be other factors that contribute to peak broadening, such as microstrain, stacking faults, and particle size distribution. Therefore, the WH plot should be used as a qualitative tool to estimate the crystallite size and strain, and other complementary techniques should be used to confirm the results.

In conclusion, the Williamson-Hall plot is a useful tool for calculating the crystallite size and strain from XRD data. The slope of the WH plot provides the crystallite size, while the y-intercept provides the strain. These parameters can provide important information about the crystal structure and properties, and can be used in various fields, such as materials science, chemistry, and physics.