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GATE 2023 PH – Question 45

Solid State Physics · diffraction methods for structure determination · 2 marks · Multiple choice

A neutron beam with a wave vector k⃗ and an energy 20.4 meV diffracts from a crystal with an outgoing wave vector k⃗ ′. One of the diffraction peaks is observed for the reciprocal lattice vector G of magnitude 3.14 Å −1 . What is the diffraction angle in degrees (rounded off to the nearest integer) that k⃗ makes with the plane? (Use mass of neutron = 1.67 × 10−27 Kg)

  1. 15
  2. 30
  3. 45
  4. 60

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Correct answer: (B) 30

Explanation

**Elastic (Bragg) diffraction.** The incoming and outgoing wave vectors have the same magnitude $k$, and the change of wave vector is a reciprocal lattice vector $\mathbf G=\mathbf k^\prime-\mathbf k$. The geometry gives
$$G=2k\sin\theta,$$
where $\theta$ is the angle between $\mathbf k$ and the diffracting plane.

**Wave vector of the neutron.** The kinetic energy is $E=20.4$ meV $=20.4\times10^{-3}\times1.602\times10^{-19}=3.268\times10^{-21}$ J:
$$k=\frac{\sqrt{2mE}}{\hbar}=\frac{\sqrt{2\times1.67\times10^{-27}\times3.268\times10^{-21}}}{1.0546\times10^{-34}}=3.13\times10^{10}\text{ m}^{-1}=3.13\ \text{\AA}^{-1}.$$

**Angle:**
$$\sin\theta=\frac{G}{2k}=\frac{3.14}{2\times3.13}=0.501\;\Rightarrow\;\theta\approx30^\circ\quad(\text{option B}).$$