Maximum Reflection of Ist order for the Crystal Planes (100), (110) and (111) of NaCl Crystal is obtained at 5.9°, 8.4° and 5.2° respectively. Find the crystal structure of NaCl.
\[ 2d \sin(\theta) = n \lambda \]
where:
- \(n\) is the order of diffraction (in this case, \(n = 1\) for the first-order reflection),
- \(d\) is the spacing between the crystal planes,
- \(\theta\) is the angle of diffraction,
- \(\lambda\) is the wavelength of incident X-rays.
Given the angles of diffraction for the (100), (110), and (111) planes of NaCl crystal as 5.9°, 8.4° and 5.2° respectively, we can use Bragg's Law to find the corresponding spacing between these planes.
For the (100) plane:
\[ 2d_{100} \sin(5.9^\circ) = 1 \times \lambda \]
For the (110) plane:
\[ 2d_{110} \sin(8.4^\circ) = 1 \times \lambda \]
For the (111) plane:
\[ 2d_{111} \sin(5.2^\circ) = 1 \times \lambda \]
Let's find the ratios of \(d_{100}\), \(d_{110}\), and \(d_{111}\).
\[ \frac{d_{100}}{d_{110}} \approx \frac{\sin(8.4^\circ)}{\sin(5.9^\circ)} \approx \frac{0.145}{0.099} \approx 1.46 \]
\[ \frac{d_{111}}{d_{100}} \approx \frac{\sin(5.2^\circ)}{\sin(5.9^\circ)} \approx \frac{0.090}{0.099} \approx 0.91 \]
Comparing these ratios to the expected ratios for a face-centered cubic (FCC) structure:
\[ \frac{d_{111}}{d_{110}} = \sqrt{3} : \sqrt{2} : 1 \approx 1.73 : 1.41 : 1 \]
The calculated ratios (\(1.46\) and \(0.91\)) are close to the expected ratios for an FCC structure. Therefore, based on these ratios, it is consistent with NaCl having a face-centered cubic (FCC) crystal structure.
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