Scherrer Equation Crystallite Size Calculator
Find the average crystallite size from X-ray diffraction peak broadening using the Scherrer equation.
🔎 What is the Scherrer Equation?
The Scherrer equation estimates the average size of coherently diffracting crystalline domains, called crystallites, from the broadening of an X-ray diffraction peak. It is given by D = Kλ/(βcosθ), where D is average crystallite size, K is a dimensionless shape factor, λ is the X-ray wavelength, β is the full width at half maximum (FWHM) of the diffraction peak, and θ is the Bragg angle at which that peak occurs.
Materials scientists and nanotechnology researchers use the Scherrer equation as a fast, routine way to estimate crystallite size directly from a powder XRD pattern, without needing electron microscopy. It is especially widely used for nanocrystalline materials, catalysts, thin films, and pharmaceutical powders, anywhere a quick size estimate from an already-collected diffraction pattern is valuable.
A common point of confusion is treating the Scherrer result as the true physical particle size. It specifically measures the coherently diffracting domain (crystallite), which can be smaller than the observable particle if a particle is made of multiple crystalline grains, and the calculated value only reflects size-related broadening, real samples often have additional broadening from microstrain that this simple form doesn't separate out.
This calculator computes D = Kλ/(βcosθ) from your shape factor, wavelength, peak broadening, and Bragg angle, showing the result in both nanometers and angstroms, and plots how crystallite size changes across a range of peak broadening values, illustrating the equation's inverse relationship directly.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Nanocrystalline Powder, Cu-Kα
K = 0.9, λ = 1.5406 Å, β = 0.5°, θ = 15°
Example 2 — Sharper Peak, Larger Crystallites
K = 0.9, λ = 1.5406 Å, β = 0.3°, θ = 10°
Example 3 — Broader Peak, Smaller Crystallites
K = 0.94 (cubic shape factor), λ = 1.5406 Å, β = 1.0°, θ = 20°
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Scherrer equation used for?
The Scherrer equation, D = K*lambda/(beta*cos theta), estimates the average size of coherently diffracting crystalline domains (crystallites) from the broadening of an X-ray diffraction peak. Smaller crystallites produce broader peaks, and the Scherrer equation quantifies that relationship.
What is the formula for the Scherrer equation?
D = K*lambda/(beta*cos theta), where D is the average crystallite size, K is a dimensionless shape factor (typically 0.9), lambda is the X-ray wavelength, beta is the full width at half maximum (FWHM) of the diffraction peak in radians, and theta is the Bragg angle.
What value of K should I use in the Scherrer equation?
K = 0.9 is the standard default for roughly spherical crystallites when beta is measured as the full width at half maximum (FWHM). Other common values range from about 0.62 to 2.08 depending on the assumed crystallite shape and exactly how the peak width is defined, always report which K value was used.
Why does peak broadening (beta) indicate smaller crystallite size?
A perfect, infinitely large crystal would produce an infinitely sharp diffraction peak. As crystallite size shrinks, there are fewer repeating lattice planes contributing to constructive interference, which broadens the diffraction peak, smaller crystallites always produce measurably broader peaks.
What is the difference between crystallite size and particle size?
Crystallite size, what the Scherrer equation measures, is the size of a single coherently diffracting crystalline domain. A particle observed under a microscope can be made up of multiple crystallites (grains), so particle size measured by microscopy or light scattering is often larger than the crystallite size measured by XRD line broadening.
What units does this calculator use?
X-ray wavelength is entered in angstroms (Å), the standard unit for diffraction, and peak broadening (beta) and Bragg angle (theta) are entered in degrees, converted internally to radians. The crystallite size result is shown in both nanometers (nm) and angstroms.
Why must instrumental broadening be subtracted from beta?
The diffractometer itself adds a small amount of peak broadening regardless of the sample, from optics, slit width, and detector resolution. This instrumental contribution must be measured on a standard reference material with large, strain-free crystallites and subtracted from the observed peak width before using beta in the Scherrer equation, otherwise crystallite size will be systematically underestimated.
Does the Scherrer equation account for microstrain?
No, the basic Scherrer equation used here assumes all peak broadening is due to crystallite size alone. Real samples often also have microstrain (lattice distortion) contributing to peak broadening, more advanced methods like the Williamson-Hall plot separate size and strain contributions using multiple diffraction peaks.
What size range is the Scherrer equation valid for?
The Scherrer equation works reliably for crystallite sizes roughly between 2 and 100-200 nanometers. Below this range, peaks become too broad and diffuse to measure precisely; above it, size-related broadening becomes too small to distinguish from the diffractometer's own instrumental peak width.
Why is Cu-Kalpha (1.5406 angstroms) the default wavelength?
Copper-target X-ray tubes producing Cu-Kalpha radiation at 1.5406 angstroms are the most common laboratory X-ray source for powder diffraction, since copper is inexpensive, produces strong characteristic radiation, and the resulting diffraction angles are convenient for most common crystal structures and detector geometries.