QB_PH_SEC7_007: Crystal Structure & Lattice Dynamics

At very low temperatures ($T \ll \Theta_D$), the lattice heat capacity $C_v$ of a three-dimensional crystalline solid varies with temperature $T$ as

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Debye Acoustic Phonon Density of States**: In 3D with linear dispersion $\omega = v_s k$: $D(\omega) = \frac{3V \omega^2}{2\pi^2 v_s^3}$ 2. **Thermal Energy Integral**: $U(T) = \int_0^{\omega_D} \frac{\hbar\omega D(\omega)}{e^{\hbar\omega / k_B T} - 1} d\omega$ Substituting $x = \hbar\omega / (k_B T)$, for $T \ll \Theta_D$ the upper limit extends to $\infty$: $U(T) \propto T^4 \int_0^\infty \frac{x^3}{e^x - 1} dx \propto T^4$ 3. **Heat Capacity**: $C_v = \frac{\partial U}{\partial T} \propto T^3$ 4. **Conclusion**: $C_v \propto T^3$ (Debye $T^3$ Law). Option (C) is correct.