In XRD, a peak at 2θ = 38.2° had a full width at half maximum (after removing instrumental broadening) of 0.40°. Using the Scherrer equation D = Kλ/(β cosθ) with K = 0.9 and λ = 0.15406 nm, which value is closest to the crystallite size?

Answer: C. About 21 nm

Convert the width to radians, β = 0.40 × π/180 ≈ 6.98×10⁻³ rad, and use half of 2θ, 19.1° (cosθ ≈ 0.945), for θ. D = 0.9 × 0.15406/(6.98×10⁻³ × 0.945) ≈ 21 nm. About 0.37 nm substitutes β as 0.40 in degrees, and about 25 nm uses the cosine of 2θ, cos38.2°, instead of θ. What the Scherrer equation gives is not the grain size itself but the size of the crystallites that diffract coherently. Nonuniform strain also broadens peaks, so strictly the two are separated by methods such as the Williamson–Hall method. When crystallites become large, the broadening is too small to determine the size accurately.