Question

The lattice constant of NaCl crystal is $0.563$ nm. X-rays of wavelength $0.141$ nm are diffracted by this crystal. The angle at which the first order maximum occurs is .......... degrees. (Specify answer up to two digits after decimal.)

Hint:

Identify the correct crystal plane before applying Bragg's law.

Answer 1

Correct Answer: 12

Step 1: Use Bragg's law.
$n\lambda = 2d\sin\theta$. For first order, $n = 1$.

Step 2: Find interplanar spacing.
For NaCl (fcc), first diffraction peak is from (111):
$d_{111} = \frac{a}{\sqrt{3}} = \frac{0.563}{1.732} = 0.325\,\text{nm}$.

Step 3: Substitute into Bragg's equation.
$\sin\theta = \frac{\lambda}{2d} = \frac{0.141}{2 \times 0.325} = 0.2167$.

Step 4: Convert to degrees.
$\theta = \sin^{-1}(0.2167) = 12.47^\circ$.
But for NaCl's first observable peak (200 reflection), spacing is $d = a/2 = 0.2815$ nm, giving:
$\theta = \sin^{-1}\left(\frac{0.141}{2 \times 0.2815}\right) = 7.21^\circ$.