QB_PH_SEC7_017: Crystal Structure & Lattice Dynamics

In the Einstein model of lattice heat capacity, the oscillators are assumed to be independent and vibrate with a single characteristic frequency $\omega_E$. At high temperatures ($T \gg \Theta_E$), the molar heat capacity approaches

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Einstein Internal Energy for 1 Mole ($N_A$ atoms, $3N_A$ modes)**: $U = 3 N_A \frac{\hbar\omega_E}{e^{\hbar\omega_E / k_B T} - 1}$ 2. **Heat Capacity Derivation**: $C_v = \frac{dU}{dT} = 3 N_A k_B \left(\frac{\hbar\omega_E}{k_B T}\right)^2 \frac{e^{\hbar\omega_E / k_B T}}{\left(e^{\hbar\omega_E / k_B T} - 1\right)^2} = 3R \left(\frac{\Theta_E}{T}\right)^2 \frac{e^{\Theta_E/T}}{(e^{\Theta_E/T}-1)^2}$ 3. **High Temperature Limit ($T \gg \Theta_E \implies x = \Theta_E/T \ll 1$)**: $e^x \approx 1 + x, \quad e^x - 1 \approx x$ $C_v \approx 3R \cdot x^2 \frac{1 + x}{x^2} \approx 3R$ 4. **Conclusion**: Approaches classical Dulong-Petit value $3R$. Option (A) is correct.