Dislocation Density from FWHM: XRD Rocking-Curve Formula Review

Answer

Determine a density by fitting a model appropriate to the material and reflection, after accounting for instrumental broadening. A rocking-curve FWHM alone is insufficient. The question contains no specimen data, so the answer is a method and its limits, not a numerical density.

1. Specify what was measured

Record the material, crystal orientation, reflection, wavelength, optics, and scan axis. An $\omega$ rocking curve probes angular orientation spread; a coupled $2\theta$ scan has a different interpretation. Their FWHMs are not interchangeable. Select reflections sensitive to the dislocation components of interest and determine the appropriate Burgers-vector magnitude $b$ from the crystal and defect type.

2. Separate instrumental and specimen contributions

Fit the peak and determine the instrument response with a suitable reference and matching geometry. For purely Gaussian profiles, widths combine in quadrature:

$$\beta_{\mathrm{sample}}=\sqrt{\beta_{\mathrm{obs}}^2-\beta_{\mathrm{inst}}^2}.$$

For purely Lorentzian profiles, the corresponding widths subtract linearly. A Voigt-type profile needs a consistent component or convolution treatment. Do not use the Gaussian correction merely because FWHMs are available. If specimen broadening cannot be resolved from the instrument response, report that limit rather than forcing a density.

3. Apply a justified density model

A commonly used threading-dislocation estimate in nitride-film rocking-curve analysis is

$$\rho\approx\frac{\beta^2}{4.35b^2}.$$

Here $\beta$ is the relevant angular width in radians, $b$ is the appropriate Burgers-vector magnitude, and the numerical factor belongs to the assumed dislocation model. The cited Scientific Reports study uses separate symmetric and asymmetric reflections for different defect components. This is an estimate under that model; it is not valid for every material or for arbitrary broadening. Other contributions, correlations among dislocations, and the treatment of tilt and twist can change the interpretation.

Convert before squaring:

$$\beta_{\mathrm{rad}}=\beta_{\mathrm{deg}}\frac{\pi}{180} =\beta_{\mathrm{arcsec}}\frac{\pi}{180\times3600}.$$

Using $b$ in metres yields $\mathrm{m}^{-2}$; using centimetres yields $\mathrm{cm}^{-2}$. Neither produces $\mathrm{cm}^{-1}$. The cited article contains a later density-unit typo; dimensional analysis of its equation fixes the unit.

4. Keep crystallite size separate

The Scherrer relation

$$D=\frac{K\lambda}{\beta_{\mathrm{size}}\cos\theta}$$

estimates a coherent diffraction-domain size when the breadth is attributable to finite size. Here $\beta_{\mathrm{size}}$ is the size contribution in a $2\theta$ profile, in radians, and $\theta$ is the Bragg angle. $K$ depends on shape and breadth convention. $D$ is not automatically a microscopy grain diameter. Substituting a rocking-curve width into this equation and reporting $1/D^2$ does not measure dislocation density; the reciprocal square of a length has the right dimensions but does not establish the defect physics.

Report the model, fitted widths, instrument treatment, reflections, $b$, units, and uncertainty. Compare with independent microscopy when available. Without those inputs, no unique specimen density follows from the original question.