XRD Peak Widths: Units, Model Choice, and Uncertainty

A diffraction peak width is an observable. A coherent-domain size or dislocation density is inferred from it using a physical model. Two calculations can have the same output units while describing different physics.

Identify the angular coordinate first

An $\omega$ rocking curve measures intensity while the specimen is rocked near a chosen reflection. A radial $2\theta$ profile measures a different direction in diffraction space. Keep the scan axis and reflection with every reported FWHM. A width measured in degrees must be converted to radians for a model written in radians; a width in arcseconds needs a further factor of 3600.

Instrument correction depends on line shape

A measured profile combines instrument response and specimen broadening. If both profiles are Gaussian, variances add, which also makes squared FWHMs additive. For Lorentzian profiles, FWHMs add. Mixed profiles require their component or convolution model. These are alternative assumptions, not two formulas to average.

As a new mathematical illustration, assume Gaussian widths of $0.18^\circ$ observed and $0.10^\circ$ instrumental. Then

$$\beta_s=\sqrt{0.18^2-0.10^2}=0.1497^\circ\approx0.002612\ \mathrm{rad}.$$

Under a hypothetical model $\rho=\beta_s^2/(4.35b^2)$ with an independently specified $b=0.50$ nm,

$$\rho\approx\frac{0.002612^2}{4.35(0.50\times10^{-9})^2} \approx6.3\times10^{12}\ \mathrm{m}^{-2}.$$

These invented inputs illustrate the calculation only; they establish no material’s actual defect density. The coefficient 4.35 is used in specific threading-dislocation models and must be justified for a real specimen.

Why a narrow denominator or small correction matters

For $\rho=\beta^2/(Cb^2)$ with fixed $C$ and $b$, doubling the width multiplies the estimate by four. A small fractional width error contributes approximately twice that fractional error to the density. This follows directly by differentiating the square law; it does not include uncertainty in the physical model.

When observed and instrumental widths nearly coincide, their squared difference is small. Small errors in either measurement can then cause a large relative error in the inferred specimen width. An unresolved width should be reported as a resolution limitation, not as proof of zero dislocations.

Distinguish length from defect density

Scherrer analysis relates size broadening to a coherent-domain length. A microscopy grain can contain multiple coherent domains. Taking the reciprocal square of the Scherrer length produces inverse-area units, but that operation alone has not counted dislocations or fitted their strain fields. Model selection must come before numerical substitution.