Line Broadening

Powder peaks get wider when the crystallites get smaller — and also when the lattice spacings vary. Give the positions and widths you measured and the page takes the instrument out, applies Scherrer to each peak, and then separates size from strain the only way a single peak cannot: by how the broadening grows with angle.

You supply: A wavelength, your instrument’s own width, and a list of peaks — each one a 2θ and a full width at half maximum in degrees. Three peaks are the minimum for a Williamson–Hall fit and they are worth more the more widely they are spread.
Reading it: Every answer comes in two versions, because taking the instrument’s width out of a measured width depends on the peak shape and a real peak is between the two limits: the answer is inside the bracket, not at either end. L is a coherently diffracting domain and not a particle size, K = 0.9 is a convention worth about ten per cent on its own, and the instrumental width has to be measured on a standard — the default here is a plausible diffractometer and is not yours.

Worked examples: size only, simulated at 15 nm · strain only, simulated at 0.30% · both at once, simulated at 25 nm and 0.15%

Input

The measurement

Wavelength λ

Å

Instrumental width

° 2θ

Scherrer constant K

The instrumental width is measured on a standard that broadens nothing of its own — LaB₆ or silicon. The default is this site’s own assumed 0.08°, which is a plausible laboratory diffractometer and is not yours.

The peaks

One peak per line: 2θ, then its full width at half maximum, both in degrees

Anything after a # is ignored, so you can keep a header row or label your peaks. Use a dot for the decimal point — a comma separates the two numbers on a line.

Results

Instrumental width is needed and was left empty.

Scherrer constant K is needed and was left empty.

Where this comes from

X-ray line broadening from filed aluminium and wolfram
G. K. Williamson and W. H. Hall, Acta Metall. 1953, 1, 22–31 · doi:10.1016/0001-6160(53)90006-6
The separation this page’s plot performs, in the paper that introduced it. The two contributions are told apart by their angular dependence alone — which is why several peaks are needed and why one is not enough, whatever its width.
The diffraction of X rays by distorted crystal aggregates - I
A. R. Stokes and A. J. C. Wilson, Proc. Phys. Soc. 1944, 56, 174–181 · doi:10.1088/0959-5309/56/3/303
Where the strain term 4ε tan θ comes from, and where L is defined as a mean column length rather than a particle size — the distinction the table of traps on this page spends the most words on.