Scherrer Equation Calculator
Calculate crystallite size from X-ray diffraction peak broadening using the Scherrer formula
Input Parameters
⚡ Auto-UpdateK depends on crystallite shape and peak profile.
β must correspond to the same peak as θ below. Use FWHM, not integral breadth.
Measure using a strain-free standard (LaB₆, Si). Quadratic: βsize = √(βobs² − βinstr²)
Uncertainty: Propagation includes βobs only. λ and θ treated as exact.
Results
Understanding the Scherrer Equation
The Scherrer equation estimates crystallite size from XRD peak broadening. First derived by Paul Scherrer in 1918, it relates diffraction peak width to domain size.
Key distinction: The Scherrer equation yields crystallite size—the coherently diffracting domain—which may be smaller than particle size.
Instrumental Correction
Measured peak width includes instrumental factors. For accuracy, subtract βinstr (from a standard like LaB₆) using quadratic subtraction.
Warning: If βobs ≈ βinstr, crystallites are likely >100 nm and Scherrer analysis becomes unreliable.
Limitations
- Micro-strain also causes broadening—consider Williamson-Hall for strained samples
- Unreliable above ~100 nm (peaks too sharp) or below ~1 nm (periodicity breaks down)
- Shape factor K varies with crystallite geometry
References
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