Engineering & Science
Debye-Scherrer XRD Crystallite Size & Peak FWHM Broadening Formula
Calculate mean nanocrystalline domain size D = (K * lambda) / (RADIANS(SQRT(FWHM_obs^2 - FWHM_inst^2)) * COS(RADIANS(2*theta / 2))) in nanometers from powder X-ray diffraction peaks.
Size nanocatalysts and thin-film grains from XRD Bragg peak broadening!
In a bulk single crystal, thousands of atomic planes destructively cancel X-rays everywhere except right at the exact Bragg angle theta—producing needle-sharp peaks! When coherent crystallite domains shrink below ~100 nm in battery cathodes, TiO2 photocatalysts, or sputtered thin films, incomplete destructive interference broadens the diffraction peak's Full Width at Half Maximum (FWHM)! Paul Scherrer (1918) derived the crystallite size law: D = (K * lambda) / (beta_rad * COS(theta)), where shape factor K = 0.90 (spherical grains), lambda = 0.15406 nm (Cu K-alpha X-rays), and theta = (2*theta) / 2!
Crucial lab step: diffractometers contribute their own instrumental slit width FWHM_inst (measured on a LaB6 or Si standard, e.g., 0.12°), which must be subtracted in quadrature for Gaussian peaks: beta_deg = SQRT(FWHM_obs^2 - FWHM_inst^2) before converting beta_deg to radians! At 2*theta = 43.30° (Cu (111) peak), an observed peak width of 0.85° (corrected beta = 0.8415°) corresponds to D = 10.16 nm nanocrystallites—growing to 19.71 nm after annealing narrows FWHM_obs to 0.45°!
| Sample Condition (2*theta = 43.30°, Cu K-alpha) | Observed vs Inst FWHM | Corrected Width beta_deg | Crystallite Size D (nm) |
|---|---|---|---|
| As-Synthesized Nanopowder | 0.85° vs 0.12° | 0.8415° | 10.16 nm |
| Annealed Coarsened Powder | 0.45° vs 0.12° | 0.4337° | 19.71 nm |
| Peak Narrower Than Instrument! | 0.10° <= 0.12° | Invalid Broadening! | "Observed FWHM <= Instrumental Width" |
Scherrer Coherent Crystallite Domain Size D (Nanometers, nm)
Subtracts instrumental FWHM B5 from observed FWHM B4 in quadrature, converts degrees to radians, and divides K * lambda by (beta_rad * COS(theta_rad)).
=IF(OR(B2 <= 0, B3 <= 0, B5 < 0, B6 <= 0, B6 >= 180), "Invalid XRD Inputs", IF(B4 <= B5, "Observed FWHM <= Instrumental Width", ROUND((B2 * B3) / (RADIANS(SQRT((B4 ^ 2) - (B5 ^ 2))) * COS(RADIANS(B6 / 2))), 2)))Quadrature-Corrected Sample Broadening beta_deg = SQRT(FWHM_obs^2 - FWHM_inst^2)
Isolates intrinsic nanocrystalline broadening in degrees prior to Scherrer evaluation.
=IF(B4 <= B5, "Observed FWHM <= Instrumental Width", ROUND(SQRT((B4 ^ 2) - (B5 ^ 2)), 4))Try the example
Use the original values or change them to test your understanding.