Scherrer Equation Calculator
Convert XRD peak broadening into crystallite size, with instrumental correction and Williamson-Hall strain separation.
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What it does
This calculator applies the Scherrer equation to powder X-ray diffraction data, converting the broadening of a diffraction peak into a crystallite size. Enter a peak position and its full width at half maximum, optionally subtract the instrumental broadening measured on a standard, and read the size of the coherently diffracting domain in nanometres. A third mode performs a Williamson-Hall analysis across several peaks to separate crystallite size from microstrain. It is aimed at materials researchers, battery and catalyst scientists, and students who need a defensible number rather than a single formula pasted into a spreadsheet.
How it works
The Scherrer equation is D = K x lambda / (beta x cos theta), where D is the size of the coherently diffracting domain in the same unit as the wavelength, K is a dimensionless shape factor, lambda is the X-ray wavelength, beta is the width of the diffraction peak of the sample itself in radians, and theta is the Bragg angle. Three details decide whether the answer is right. First, beta must be in radians: laboratory software reports FWHM in degrees, and substituting degrees inflates D by a factor of 57.3 while still producing a plausible looking nanometre number. Second, theta is the half angle: the measured quantity is 2 theta, so halve it before taking the cosine. Third, beta must be the contribution of the sample alone, so the instrumental broadening has to be removed. If both profiles are approximately Gaussian their variances add and beta = sqrt(beta_obs^2 - beta_inst^2); if they are Lorentzian the widths subtract directly. The two models typically differ by 5 to 15 percent, and more when the sample peak is narrow. The shape factor K is about 0.89 for spherical crystallites, 0.83 for cubes, and 0.9 is a common compromise for randomly oriented powders. The wavelength depends on the source: Cu K alpha is 0.15406 nm, Co K alpha 0.17902 nm, and Mo K alpha 0.071073 nm.
Worked example
A Cu K alpha pattern (lambda = 0.15406 nm) shows a peak at 2 theta = 21.50 degrees with an observed FWHM of 0.318 degrees. An LaB6 standard measured under the same conditions gives an instrumental FWHM of 0.080 degrees. Halving the angle gives theta = 10.75 degrees, so cos theta = 0.98245. Converting to radians, beta_obs = 0.00555 rad and beta_inst = 0.00140 rad. Assuming Gaussian profiles, beta = sqrt(0.00555^2 - 0.00140^2) = 0.00537 rad. Then D = 0.9 x 0.15406 / (0.00537 x 0.98245) = 26.3 nm. Leaving the instrumental term out would have given 25.4 nm, so the correction matters at the few percent level here. Now feed eight peaks of the same phase into the Williamson-Hall mode and the picture changes: the apparent single-peak size drifts from 25.4 nm at low angle down to 18.0 nm at high angle, the classic signature of microstrain. The fitted line has an intercept of 4.335e-3 and a slope of 1.499e-3, so D = 0.9 x 0.15406 / 4.335e-3 = 32.0 nm and the microstrain is 1.50e-3. The true crystallite size is 32 nm, about a quarter larger than any single peak suggested.
When to use it
Use the single peak mode for a quick estimate when only one well-resolved reflection is available, or to compare a series of samples measured under identical conditions, where systematic errors largely cancel. Use the reverse mode before an experiment: give it the size you expect and it returns the peak width you would need in order to resolve it, which tells you whether the measurement is feasible on your instrument at all. Use the Williamson-Hall mode whenever the sample has been heat treated, milled or otherwise strained, because in that case single-peak Scherrer sizes are not merely imprecise but systematically wrong. The honest limits of the method are worth stating. The equation returns the size of the coherently diffracting domain, not the particle size, so a polycrystalline particle gives a number smaller than its physical dimensions, sometimes much smaller. It is reliable roughly between 2 and 100 nm, degrades badly above 200 nm where the broadening falls below instrument resolution, and assumes a narrow monodisperse size distribution and isotropic strain. It knows nothing about the shape of the size distribution, so a bimodal powder yields a meaningless average. For large crystallites or full structural detail, Rietveld refinement or transmission electron microscopy is the right tool; Scherrer remains the fastest way to put a first number on a nanocrystalline sample.
FAQ
- What is the difference between crystallite size and particle size?
- The Scherrer equation returns the size of the coherently diffracting domain, also called a crystallite: the region over which the crystal lattice is continuous and in phase. A particle is a physically separate object that may consist of many crystallites joined at grain boundaries, stacking faults or twins. Where the two differ, the crystallite size is always the smaller number. A 200 nm polycrystalline particle built from 30 nm crystallites will report about 30 nm. This is the most common source of confusion when the result is compared with scanning electron microscopy, which sees particle outlines, or with laser diffraction, which sees agglomerates. If you need the particle size, use microscopy. If you need to know how coherently a nanomaterial is ordered, which controls peak broadening and often catalytic and electrochemical behaviour, the Scherrer number is the informative one.
- Why must the FWHM be converted to radians?
- Because theta in the cosine term is an angle, and the product beta times cos theta is only meaningful when both are in the same unit system. Laboratory software reports peak widths in degrees because that is convenient for plotting, but the Scherrer equation as written requires beta in radians. Substituting degrees inflates D by 57.2958, the number of degrees in one radian. The failure is hard to catch by eye because the result is still a plausible nanometre figure: a 40 nm crystallite becomes 2348 nm, which many users simply read as a large crystallite or a bad sample. Good implementations convert internally, as this calculator does. When you compute by hand or in a spreadsheet, you must do it yourself.
- Why is my crystallite size smaller than the particle size from microscopy?
- Because they measure different things. Scherrer weights only the volume that diffracts coherently, so anything that interrupts lattice continuity, such as a grain boundary, a stacking fault, a twin or a high density of dislocations, removes that volume from the calculation. Microscopy counts the entire particle outline, including amorphous surface layers, which are common on nanoparticles and can be one or two nanometres thick. The comparison is only meaningful when the size distribution is also examined: Scherrer returns a volume-weighted average dominated by the most ordered regions, so a powder with a broad distribution will not match a number counted from a micrograph. If a well-crystallised sample disagrees by more than about a factor of two, suspect instrumental broadening, because an underestimated correction artificially shrinks the reported size.
- Can I use a single peak, or do I need a Williamson-Hall analysis?
- A single peak gives a usable estimate only when the sample is strain-free, which is rare for anything that has been milled, heat treated or deposited as a thin film. Size broadening and strain broadening have different angular dependences: size broadening goes as one over cos theta and dominates at low angle, while strain broadening goes as tan theta and dominates at high angle. A single measurement cannot separate two contributions with different shapes, so it returns a blend. The Williamson-Hall method measures several peaks across a wide angular range and fits beta cos theta against 4 sin theta, which linearises both terms: the intercept is K lambda over D and the slope is the microstrain. If the peaks of a sample give apparent single-peak sizes that drift with angle, strain is present and only the multi-peak fit is trustworthy. Use the single peak mode for rapid cross-sample comparison, and the Williamson-Hall mode whenever the number is going into a paper.