Diffraction by Organized Layers — Question 9

Question 9

Instrumental broadening must be removed before estimating crystallite size

Use the supplied Scherrer model $T=K\lambda/(B\cos\theta)$, where B is sample FWHM in $2\theta$ radians and $K=0.90$. Assume Gaussian widths, so $B^2=B_{\mathrm{measured}}^2-B_{\mathrm{instrument}}^2$. A peak at $2\theta=60^\circ$ has measured FWHM $0.200^\circ$, instrument FWHM $0.120^\circ$, and $\lambda=0.154\,\mathrm{nm}$. Neglect strain broadening.

1. Find the corrected width and estimated coherent crystallite thickness.

2. Calculate the size estimate obtained if instrumental broadening were ignored and explain the direction of the bias.

Question 9 - Solution

Instrumental broadening must be removed before estimating crystallite size

(a) Method and calculation

$$B=\sqrt{0.200^2-0.120^2}\,\mathrm{deg}=0.160^\circ=0.00279253\,\mathrm{rad}.$$$$\boxed{T=\frac{0.90(0.154\,\mathrm{nm})}{0.00279253\cos30^\circ}=57.3106\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$B_{\mathrm{measured}}=0.00349066\,\mathrm{rad}\Longrightarrow\boxed{T_{\mathrm{uncorrected}}=45.8485\,\mathrm{nm}}.$$$$\boxed{T_{\mathrm{uncorrected}}/T=B/B_{\mathrm{measured}}=0.80}.$$

Physical check

Assigning instrumental width to the sample makes the coherent domain appear too small. This model estimates coherent domain size, not necessarily the full particle size.