4. Crystallite Size Calculation
Learn how the Scherrer equation estimates nanocrystalline sizes. Read about shape factor choices and instrument calibrations.
Guides List
The Physics of Peak Broadening
An infinite, perfect crystal lattice diffracted under monochromatic light creates delta-function reflections. However, real materials feature finite crystallite sizes and defects that disrupt this perfect symmetry, causing the diffraction peaks to broaden.
The Scherrer Formula
The average size of a single coherent crystalline domain is calculated using the Scherrer formula:
Where:
- D: The mean crystallite size (in Angstroms \AA).
- K: The shape factor coefficient (typically $0.9$ for spherical grains, but ranges from $0.62$ to $2.08$ depending on lattice geometry).
- λ: The incident X-ray source wavelength (typically Cu $K\alpha_1 = 1.5406$ \AA).
- β_crystallite: The peak width (FWHM) in radians, corrected for instrument broadening.
- θ: The Bragg angle (half the $2\theta$ peak center).
Subtracting Instrument Broadening
Before applying the Scherrer equation, you must subtract the broadening contribution from the instrument itself. This is done by measuring a standard reference material (like LaB6 or NIST Silicon) to determine the instrument's FWHM curve (βinst).
You can then subtract this instrument broadening using one of these options:
- Linear Subtraction (Lorentzian Profiles): Used when peak profiles are purely Lorentzian:
β_crystallite = β_measured - β_instrument - Quadratic Subtraction (Gaussian Profiles): Used when peak profiles are purely Gaussian:
β_crystallite = √(β_measured² - β_instrument²)
Limitations and Guidelines
The Scherrer equation is an estimation tool with specific limits. It is only reliable for crystallite sizes under 100 nm. For domains larger than 100 nm, peak broadening is too small to distinguish from instrument limits. It also assumes the sample is free from lattice strains, which must be verified using the Williamson-Hall method.