BCS theory basics (introductory)
Solid state phenomena · Physics
Study notes
Worked example: Niobium has Tc = 9.2 K. Estimate its BCS energy gap and the photon frequency needed to break a Cooper pair. Step 1: BCS ratio 2Δ = 3.53·k_B·T_c = 3.53 × 1.381×10⁻²³ × 9.2 = 4.49×10⁻²² J. Step 2: In eV: 4.49×10⁻²² / 1.602×10⁻¹⁹ ≈ 2.8 meV. Step 3: Breaking a pair needs photon energy ≥ 2Δ: frequency ν = 2Δ/h = 4.49×10⁻²²/6.626×10⁻³⁴ ≈ 6.8×10¹¹ Hz. Step 4: That's ~0.68 THz (far-infrared/microwave boundary) — thermal photons at 9 K have energy kT ≈ 0.78 meV << 2.8 meV, so pairs survive. Conclusion: the meV gap is tiny yet decisive — below Tc, thermal agitation cannot break pairs, so the condensate flows without resistance.