the Scherrer equation

The Scherrer equation is the simple formula that links how fat a diffraction peak is to how small the crystals are. The intuition: diffraction is sharp only because vast numbers of parallel planes interfere and cancel almost perfectly at any angle just off the Bragg angle. Shrink the crystallite so it has only a few dozen planes, and that cancellation becomes incomplete, so the peak spreads out. Broad peak means tiny crystal.

The formula is L = K lambda / (beta cos theta). Here L is the mean crystallite size, K is a shape constant near 0.9, lambda is the X-ray wavelength, beta is the extra peak breadth (full width at half maximum, in radians) after you have removed the instrument's own broadening, and theta is the Bragg angle. Worked example: take lambda = 1.5406 angstrom, a measured beta = 0.006 radian (about 0.34 degrees), and theta = 20 degrees, so cos theta = 0.94. Then L = (0.9 x 1.5406)/(0.006 x 0.94) = about 246 angstrom, roughly 25 nanometres.

This gives a fast, nondestructive estimate of nanocrystallite size straight from a single peak, which is why it is everywhere in nanomaterials work. Be candid about its limits, though. It assumes all the broadening is from size, ignoring strain (which also broadens peaks) — so use it only when strain is negligible or has been separated out. The constant K depends on crystallite shape and on exactly how you define the width. It gives a volume-weighted mean, not a full size distribution. And above roughly 100 to 200 nanometres the size broadening is smaller than the instrument's, so Scherrer simply stops being able to tell.