Crystallite size calculation from XRD diffraction data

Introduction:

Crystalline materials have a periodic arrangement of atoms in a lattice structure, and the size of the crystallites, which are regions of the lattice with similar orientation, can be an important factor in determining the material properties. X-ray diffraction (XRD) is a powerful technique for studying crystalline materials, providing information about the crystal structure, orientation, and size of the crystallites. In this article, we will discuss the theory and methods used to calculate crystallite size from XRD data.

Theory:

The crystallite size can be calculated using the Scherrer equation, which relates the line broadening (peak width) in the XRD pattern to the crystallite size. The equation is:

D = Kλ / (βcosθ)

where D is the crystallite size, K is a dimensionless shape factor (typically between 0.9 and 1.2), λ is the X-ray wavelength, β is the peak width at half maximum (in radians), and θ is the Bragg angle. The Bragg angle is given by:

θ = 2θ - 2δ

where θ is the diffraction angle, and δ is the angle of incidence.

The Scherrer equation assumes that the crystallites are small and randomly oriented, and that the line broadening is due to size effects alone. It also assumes that the XRD peak is well-resolved and free from other contributions such as strain or defects. These assumptions are generally valid for crystallites with sizes in the range of a few nanometers to several micrometers.

Methods:

There are several methods for determining the peak width (β) in the Scherrer equation. One common method is the full width at half maximum (FWHM) method, which measures the width of the peak at half its maximum intensity. The FWHM method assumes that the peak is symmetrical, which is not always the case, and can underestimate the peak width if the peak is asymmetrical.

Another method is the integral breadth method, which integrates the area under the peak and normalizes it to the peak height. The integral breadth method is less sensitive to peak asymmetry than the FWHM method, but can be affected by background subtraction errors.

The Warren-Averbach method is another method for determining the peak width, which involves measuring the Fourier transform of the XRD pattern and fitting a Lorentzian or Gaussian function to the resulting peaks. The Warren-Averbach method is more accurate than the FWHM and integral breadth methods, but is also more time-consuming.

Regardless of the method used to determine the peak width, it is important to account for instrumental broadening, which is caused by factors such as the X-ray source, collimation, and detector. Instrumental broadening can be estimated by measuring the peak width of a reference material with a known crystallite size, such as corundum (Al2O3) or silicon (Si).

In addition to the Scherrer equation, there are other methods for calculating crystallite size from XRD data, such as the Williamson-Hall method and the size-strain plot method. The Williamson-Hall method involves plotting the peak width as a function of the sine of the Bragg angle, and fitting a straight line to the data. The intercept of the line with the x-axis gives the crystallite size, while the slope gives the strain. The size-strain plot method involves plotting the crystallite size as a function of the strain, and fitting a straight line to the data.

Applications:

The crystallite size calculated from XRD data can provide valuable information for understanding the properties and behavior of materials. For example, small crystallite sizes can lead to increased surface area, which can enhance catalytic activity and vice versa.