# Step-by-Step Solution ## Explanation of Crystallite Size and Peak Broadening Crystallite size affects the width of diffraction peaks in powder X-ray diffraction (XRD) patterns. Smaller crystallites cause broader peaks due to the uncertainty principle: as the number of diffracting planes decreases, the precision in their orientation decreases, broadening the diffraction peak. This phenomenon is known as size broadening. ## Calculation Using the Scherrer Equation The Scherrer equation relates the crystallite size (\( D \)) to the broadening of a specific XRD peak: \[ D = \frac{K \lambda}{\beta \cos\theta} \] Where: - \( D \): Crystallite size - \( K \): Shape factor (usually .9) - \( \lambda \): X-ray wavelength (given as .154 nm) - \( \beta \): Full width at half maximum (FWHM) in radians - \( \theta \): Bragg angle (half of \( 2\theta \)) ### Step 1: Convert FWHM to Radians Given FWHM (\( \beta \)) in \( 2\theta \) scale is .3°. Convert degrees to radians: \[ \beta (\text{radians}) = .3^\circ \times \frac{\pi}{180} = .00524 \text{ radians} \] ### Step 2: Calculate \( \theta \) Given \( 2\theta = 30^\circ \), so \[ \theta = \frac{30^\circ}{2} = 15^\circ \] ### Step 3: Calculate \( \cos\theta \) \[ \cos 15^\circ = .9659 \] ### Step 4: Substitute into the Scherrer Equation \[ D = \frac{.9 \times .154}{.00524 \times .9659} \] Calculate the denominator: \[ .00524 \times .9659 = .00506 \] Then: \[ D = \frac{.1386}{.00506} = 27.4~\text{nm} \] **Final Answer:** The calculated crystallite size is **27.4 nm**. --- ## Limitations of the Scherrer Equation The Scherrer equation provides only an estimate for crystallite size and has several limitations: - It assumes that peak broadening is due only to small crystallite size, whereas other factors, such as lattice strain or instrumental broadening, can also contribute. - It is most accurate for crystallite sizes below about 100 nm; for larger sizes, the peaks become too sharp for reliable measurement. - The shape factor (\( K \)) depends on the crystallite shape and may not always be exactly .9. - The equation assumes that the crystallites are free of strain and that the sample is free of defects, which is rarely the case in real materials. - Instrumental broadening must be subtracted from the observed FWHM to avoid overestimating broadening due to small size. In summary, while the Scherrer equation is a useful tool for estimating crystallite size from XRD data, its results should be interpreted with caution and supplemented with other characterization techniques for more accurate size determination.