Discussion
An X-ray diffraction pattern of cubic crystal of lattice parameter a = 3·16 Å is obtained using a monochromatic X-ray beam of wavelength 1·54 Å. The first line is obtained at θ = 20·3. The Miller indices (hkl value) of the corresponding diffracting plane is :
(Note : sin 20·3 = 0·34, sin 40·6 = 0·64)
Answer: A
The plane is (110).
From Bragg's law in first order, the spacing of the diffracting planes is
\[ \begin{aligned} d &= \frac{\lambda}{2\sin\theta} = \frac{1.54}{2 \times 0.34} \\ &= 2.265\ \text{\AA} \end{aligned} \]
For a cubic crystal \( d = a/\sqrt{h^{2} + k^{2} + l^{2}} \), so
\[ \begin{aligned} \sqrt{h^{2} + k^{2} + l^{2}} &= \frac{3.16}{2.265} \\ &= 1.395 \approx \sqrt{2} \end{aligned} \]
The only integers with \( h^{2} + k^{2} + l^{2} = 2 \) are 1, 1, 0. The other planes give different spacings: (100) gives 3.16 Å, (200) gives 1.58 Å and (220) gives 1.12 Å. A first line from (110) is what a body-centred cubic lattice produces.
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