QB_PH_SEC7_027: Crystal Structure & Lattice Dynamics

In the long-wavelength limit ($k \to 0$), the group velocity $v_g = d\omega/dk$ of acoustic phonons in a one-dimensional lattice is

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Monoatomic 1D Lattice Dispersion**: $\omega(k) = 2 \sqrt{\frac{C}{M}} \left|\sin\left(\frac{ka}{2}\right)\right|$ 2. **Long-Wavelength Limit ($k \to 0 \implies ka \ll 1$)**: $\sin\left(\frac{ka}{2}\right) \approx \frac{ka}{2}$ $\omega(k) \approx 2 \sqrt{\frac{C}{M}} \left(\frac{ka}{2}\right) = \left(a \sqrt{\frac{C}{M}}\right) k = v_s k$ 3. **Group Velocity Computation**: $v_g = \frac{d\omega}{dk} = a \sqrt{\frac{C}{M}} = v_s = \text{constant}$ 4. **Conclusion**: Constant and equal to sound velocity. Option (A) is correct.