Crystallite Size Calculator
Enter the peak position in 2θ, its full width at half maximum, the instrument broadening, the X-ray tube and the shape factor K. The calculator gives the crystallite size in nanometres using the Scherrer equation.
- Scherrer equation
- Instrument correction
- Cu, Co, Mo or Cr tubes
Enter the peak width in degrees 2θ, not radians. Instrument broadening comes from a standard such as LaB₆ or silicon.
Small crystals make broad X-ray peaks. The Scherrer equation turns peak width into size: τ = Kλ ÷ (β cos θ). A peak at 38° that is 0.3° wide after correction comes from crystallites about 28 nm across.
Crystallite size chart
Cu Kα, 2θ = 38.2°, 0.05° instrument. Search for a width.
| FWHM (°) | Size (nm) |
|---|---|
| 0.1 | 97.1 |
| 0.2 | 43.4 |
| 0.3 | 28.4 |
| 0.5 | 16.9 |
| 0.8 | 10.5 |
| 1 | 8.4 |
| 1.5 | 5.6 |
Sizes from the Scherrer equation with K = 0.9.
Peak width and size
Cu Kα, 2θ 38.2°. Slide the peak width.
The curve is the diffraction peak. The marked width is the full width at half maximum.
How to use the Scherrer equation
Width to size. Step through it.
Fit the peak
Find FWHM.
Fit the peak with a Gaussian, Lorentzian or pseudo-Voigt curve to get a reliable width.
Scherrer, Williamson-Hall or TEM
Methods. Compare them.
One peak.
Several peaks.
Direct imaging.
Electron microscopy measures particles, which can hold several crystallites, so sizes often differ.
Before reporting
XRD checks. Tick each item.
Tick each item as you go.
Report K, the wavelength and how you corrected for the instrument, so others can repeat your result.
What you get
Crystallite size. Each tile explains one result.
nm and Å.
Degrees and radians.
Half of 2θ.
From the tube.
Range check.
The method only works up to about 100 to 200 nm, where peak broadening becomes too small to measure.
Frequently Asked Questions
No question matches that search.
What is the Scherrer equation?
Crystallite size = K λ ÷ (β cos θ), where β is the peak width in radians and θ is half the peak position.
What value of K should I use?
Usually 0.9 for spherical crystallites, though 0.89 to 0.94 are also used.
Why subtract instrument broadening?
The instrument widens every peak a little, so that part must be removed before working out size.
What wavelength is Cu Kα?
1.5406 Å, or 0.15406 nm.
Is crystallite size the same as particle size?
No. A particle can contain many crystallites.
What sizes can the Scherrer equation measure?
About 2 to 100 nm reliably.
Does strain affect peak width?
Yes. Strain also broadens peaks, so Scherrer gives a lower limit to size.
Size from a single peak
Fit the peak, correct for the instrument and apply the Scherrer equation. Check several peaks to make sure strain is not misleading you.
More size calculators
Each one has its own calculator, worked examples and answers to common questions.