QB_PH_SEC7_006: Dielectric, Magnetic Properties & Superconductivity

According to the BCS theory of superconductivity, what is the ratio of the zero-temperature superconducting energy gap $2\Delta(0)$ to $k_B T_c$? (Give your answer to two decimal places)

Correct Answer

### Step-by-Step Derivation & Verification: 1. **BCS Gap Equation**: At $T = 0$, the gap $\Delta(0)$ satisfies: $\Delta(0) = 2\hbar\omega_D \exp\left(-\frac{1}{N(0)V}\right)$ 2. **Transition Temperature**: The critical temperature $T_c$ satisfies: $k_B T_c = 1.134\, \hbar\omega_D \exp\left(-\frac{1}{N(0)V}\right) = \frac{2 e^{-\gamma}}{\pi} \hbar\omega_D \exp\left(-\frac{1}{N(0)V}\right)$ 3. **Universal Ratio**: $\frac{2\Delta(0)}{k_B T_c} = \frac{4}{\frac{2 e^{-\gamma}}{\pi}} = 2\pi e^\gamma \approx 2 \times 3.14159 \times 0.5615 = 3.528 \approx 3.53$ 4. **Standard Range**: $[3.50, 3.55]$.