16.391
nm
163.91
Å
0.008727
rad
0.96936
16.391
nm
163.91
Å
0.008727
rad
0.96936
The Debye-Scherrer Crystallite Size Calculator determines the average size of crystalline domains from X-ray diffraction peak broadening. When crystallites are smaller than approximately 100 nm, diffraction peaks become measurably broader than the instrumental resolution, and this broadening is inversely proportional to crystallite size. The Debye-Scherrer equation quantifies this relationship.
This analysis is essential in nanomaterials science, catalysis, pharmaceuticals, and thin film technology where crystallite size directly impacts material properties. Nanoparticle catalysts, quantum dots, and nanostructured coatings all require accurate size characterization, and XRD line broadening provides a volume-averaged measurement complementary to microscopy techniques.
The Debye-Scherrer equation relates crystallite size to peak broadening:
$$D = \frac{K\lambda}{\beta\cos\theta}$$
Where $$D$$ is the average crystallite size (in the direction perpendicular to the diffracting planes), $$K$$ is the shape factor (dimensionless, typically 0.89–0.94), $$\lambda$$ is the X-ray wavelength, $$\beta$$ is the full width at half maximum (FWHM) of the diffraction peak in radians, and $$\theta$$ is the Bragg angle.
Important considerations:
The measured broadening must be corrected for instrumental broadening: $$\beta_{\text{sample}} = \sqrt{\beta_{\text{measured}}^2 - \beta_{\text{instrument}}^2}$$ (for Gaussian profiles) or $$\beta_{\text{sample}} = \beta_{\text{measured}} - \beta_{\text{instrument}}$$ (for Lorentzian profiles).
The shape factor K depends on crystallite shape and the definition of "size." K = 0.9 is commonly used for spherical particles with FWHM broadening. For the integral breadth method, K = 1.0 is typical.
D represents the volume-weighted average dimension of coherently diffracting domains, not necessarily the particle size. A 50 nm particle containing grain boundaries would show a crystallite size smaller than 50 nm. Values below 5 nm should be interpreted cautiously due to the extremely broad peaks involved. Sizes above 100 nm approach the resolution limit of most laboratory diffractometers, where instrumental broadening dominates.
Inputs
Results
D = (0.9 × 1.5406) / (0.00873 × cos(14.22°)) = 1.387 / 0.00846 ≈ 159 Å = 15.9 nm
Inputs
Results
Broad peak (2° FWHM) indicates very small crystallites: D ≈ 4.2 nm gold nanoparticles
The Debye-Scherrer equation D = Kλ/(β cosθ) calculates the average crystallite size from X-ray diffraction peak broadening. It was derived by Paul Scherrer in 1918 based on the idea that a finite number of diffracting planes produces a peak of finite width, with narrower peaks from more planes (larger crystallites).
K = 0.9 (or 0.89) is the most common choice for spherical crystallites using FWHM. For cubic crystallites, K ≈ 0.94. For the integral breadth definition, K = 1.0. The exact value depends on crystallite shape, size distribution, and how broadening is measured. Using K = 0.9 introduces at most 10–15% uncertainty.
Measure a standard with large crystallites (e.g., LaB6 NIST SRM 660c or well-annealed silicon) under the same conditions. The standard's peak width represents instrumental broadening (β_inst). For Gaussian profiles: β²_sample = β²_meas - β²_inst. For Lorentzian profiles: β_sample = β_meas - β_inst. Pseudo-Voigt profiles require more complex deconvolution.
FWHM (Full Width at Half Maximum) is the width of a diffraction peak measured at half its maximum intensity. It is measured in degrees 2θ and must be converted to radians for the Debye-Scherrer equation. FWHM can be determined by peak fitting (Gaussian, Lorentzian, or pseudo-Voigt functions) or direct measurement from the diffraction pattern.
Not necessarily. Crystallite size refers to coherently diffracting domains. A single particle may contain multiple crystallites separated by grain boundaries, twin boundaries, or other defects. Conversely, in a single crystal, the crystallite size equals the particle size. TEM imaging can distinguish crystallite and particle sizes.
Limitations include: (1) assumes only size broadening, no microstrain; (2) valid mainly for 2–100 nm range; (3) gives a volume-weighted average, not a size distribution; (4) shape factor uncertainty of ~15%; (5) assumes uniform, isotropic crystallites. For anisotropic or strained samples, the Williamson-Hall method is preferred.
The Williamson-Hall method separates size and strain contributions to peak broadening. It plots β cosθ versus sinθ; the intercept gives Kλ/D (size) and the slope gives 4ε (microstrain). This is more reliable than Debye-Scherrer when both size and strain broadening are present, which is common in nanocrystalline and deformed materials.
No. Amorphous materials lack long-range order and produce broad humps rather than sharp diffraction peaks. The Debye-Scherrer equation requires identifiable Bragg peaks. However, the width of the amorphous hump can give a rough estimate of short-range order correlation lengths, typically 0.5–2 nm.
The cosθ term arises from the geometry of diffraction. Peak broadening in reciprocal space is constant (independent of angle) for pure size broadening, but converting from reciprocal space (Δs) to angular space (Δ2θ) introduces the cosθ factor through the derivative of Bragg's law: Δ(2θ) = 2Δs/(cosθ).
Accuracy is typically ±20–30% due to uncertainties in K, peak shape deconvolution, and the assumption of monodisperse spherical crystallites. For relative comparisons between samples measured under identical conditions, precision is much better (±5–10%). For absolute size determination, calibration with TEM measurements is recommended.
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