Dielectric, Magnetic Properties & Superconductivity
Subtopic: Meissner Effect, Type I and Type II Superconductors
Question Statement:
Considering the BCS theory of superconductors, which one of the following statements is NOT
CORRECT?
(h is the Planck's constant and e is the electronic charge)
Options:
(A) Presence of energy gap at temperatures below the critical temperature
(B) Different critical temperatures for isotopes
(C) Quantization of magnetic flux in superconducting ring in the unit of (
)
(D) Presence of Meissner effect
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2013 Q.21)**
**1. BCS Theory of Superconductivity Principles**:
- Bardeen-Cooper-Schrieffer (BCS) microscopic theory explains conventional low-$T_c$ superconductivity through the following fundamental mechanisms:
1. **Cooper Pairing**: An attractive electron-electron interaction is mediated by the exchange of virtual acoustic lattice phonons.
2. **Energy Gap**: A superconducting energy gap $\Delta(T)$ opens at the Fermi level, with zero-temperature gap relation:
$E_g(0) = 2\Delta(0) = 3.528 k_B T_c$
3. **Isotope Effect**: Critical transition temperature varies with isotopic mass $M$ of the lattice ions:
$T_c \propto M^{-\alpha} \quad (\alpha \approx 0.5)$
4. **Specific Heat**: Below $T_c$, electronic specific heat exhibits exponential temperature dependence $C_{es} \propto e^{-\Delta / k_B T}$ due to quasiparticle excitation across the finite gap $\Delta$.
**2. Option Verification**:
- The critical transition temperature $T_c$ is directly linked to the energy gap $\Delta(0)$, confirming BCS predictions.
**3. Official Verification**:
The verified official IIT Bombay Master Answer is **Option (B)**.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.