GATE 2024 PH – Question 39
The X-ray diffraction pattern of a monatomic cubic crystal with rigid spherical atoms of radius 1.56 Å shows several Bragg reflections of which the reflection appearing at the lowest 2θ value is from (111) plane. If the wavelength of X-ray used is 0.78 Å, the Bragg angle (in 2θ, rounded off to one decimal place) corresponding to this reflection and the crystal structure, respectively, are
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Correct answer: (B) 17.6° and face centered cubic
Explanation
**Step 1: crystal structure.** For a rigid-sphere monatomic cubic crystal the lowest-angle reflection is (111) for FCC (the allowed reflections of an FCC lattice are those with all-odd or all-even indices, and the first is 111). For BCC the first allowed reflection is (110). Since the lowest reflection is (111), the crystal is **face-centred cubic**.
**Step 2: lattice constant.** In FCC the atoms touch along a face diagonal: $4r=a\sqrt2$, so
$$a=2\sqrt2\,r=2\sqrt2\times1.56=4.412\text{ Å}.$$
**Step 3: spacing of (111) planes.**
$$d_{111}=\frac{a}{\sqrt3}=\frac{4.412}{1.732}=2.547\text{ Å}.$$
**Step 4: Bragg's law** $2d\sin\theta=\lambda$:
$$\sin\theta=\frac{0.78}{2\times2.547}=0.1531\;\Rightarrow\;\theta=8.80^\circ\;\Rightarrow\;2\theta=\mathbf{17.6^\circ}.$$
Answer: 17.6° and face-centred cubic (B).