Superconductivity

Discovery and history

Year	Discovery	Material	T_c
1911	Onnes — first superconductor	Mercury	4.15 K
1933	Meissner effect observed	Tin, lead	—
1957	BCS theory explains conventional SC	—	—
1962	Josephson effect predicted	—	—
1986	Bednorz, Müller — high-T_c cuprates	LBCO	30 K
1987	Wu, Chu — first above N₂ boiling	YBa₂Cu₃O₇	93 K
1995	HgBaCaCuO discovered	HgBaCaCuO	135 K
2015	Hydrogen sulfide at high P	H₃S, 200 GPa	203 K
2019	Lanthanum superhydride	LaH₁₀, 170 GPa	250 K
2023	LK-99 claim (later debunked)	Pb-Cu-doped apatite	(not superconducting)

Key properties

Property	Description
Zero resistance	Below T_c, R = 0; current persists indefinitely
Meissner effect	Magnetic fields expelled (Type I) or pinned (Type II)
Energy gap	2Δ between superconducting and normal states; thermal blocked
Critical T (T_c)	Above T_c, normal metal; below, superconductor
Critical field (H_c)	Magnetic field above which SC destroyed
Critical current	Current density above which SC destroyed
Cooper pairs	Bound electron pairs that carry current
Coherence length	Size of Cooper pair (~ nm typical)

Type I vs Type II

Aspect	Type I	Type II
Magnetic field	Fully expelled (Meissner)	Penetrates as flux quanta (vortices)
Critical field	One H_c	H_c1 (start of penetration), H_c2 (full destruction)
Examples	Mercury, lead (pure)	Niobium, alloys, all high-T_c
High current	Limited	Can carry high currents in mixed state
Practical use	Limited (low fields only)	MRI, accelerators, etc.

JavaScript — superconductivity

// BCS energy gap (T-dependent)
function bcsGap(T_K, T_c, gap_0) {
  // 2Δ(0) ≈ 3.52 k_B T_c (BCS prediction)
  // Gap drops to zero at T_c
  if (T_K >= T_c) return 0;
  return gap_0 * Math.sqrt(1 - T_K/T_c);  // approximate
}

const k_B = 1.38e-23;
const gap_0_Pb = 3.52 * k_B * 7.2 / 2;  // Lead, T_c = 7.2 K
console.log(`Pb gap at 4 K: ${(bcsGap(4, 7.2, gap_0_Pb) / 1.602e-19 * 1e3).toFixed(2)} meV`);

// Critical magnetic field (T-dependent)
function criticalField(T_K, T_c, H_c0) {
  if (T_K >= T_c) return 0;
  return H_c0 * (1 - (T_K/T_c) * (T_K/T_c));
}

// Lead H_c0 ≈ 0.08 T
console.log(`Pb H_c at 4 K: ${criticalField(4, 7.2, 0.08).toFixed(3)} T`);

// Energy savings from superconducting transmission
function losssalessavings(P_load, V, R_normal, length) {
  // Normal: P_loss = (P/V)² × R × length
  const lossNormal = (P_load/V) ** 2 * R_normal * length;
  return lossNormal;
}

// 1 GW over 1000 km of normal cable, vs zero loss for SC
console.log(`Normal cable loss: ${(losssalessavings(1e9, 4e5, 5e-9, 1e3) / 1e6).toFixed(0)} kW per km`);
// SC: 0 (but cooling cost matters)

// Cooling cost for cryogenic SC system
function cryoCost(power_dissipated, T_cold = 77, T_hot = 300) {
  // Carnot efficiency: COP_carnot = T_cold/(T_hot-T_cold)
  // For 77K to 300K: COP ~ 0.345 (need 2.9 W cooling per W heat removed)
  const COP_real = 0.1 * 77 / (300 - 77);  // assume 10% of Carnot
  return power_dissipated / COP_real;  // electricity cost
}

console.log(`To remove 1 W at 77K: ${cryoCost(1).toFixed(0)} W input`);

// MRI superconducting magnet field
function mriFieldEnergy(B, V) {
  // U = B²·V / (2μ_0)
  return B * B * V / (2 * 4 * Math.PI * 1e-7);
}

// 3T MRI, ~1 m³ active volume
console.log(`3T MRI energy: ${(mriFieldEnergy(3, 1) / 1e6).toFixed(0)} MJ`);

Where superconductors matter

• MRI magnets. Most clinical MRI uses superconducting magnets (1.5 T or 3 T typical).
• Particle accelerators. LHC, Tevatron, ITER all use superconducting magnets.
• Maglev trains. Japan's SCMaglev uses superconducting electromagnets.
• Quantum computing. Some qubit designs (transmons) use superconducting circuits.
• SQUID detectors. Most sensitive magnetic field detectors known (10⁻¹⁵ T).
• Power transmission. Limited use; cooling cost vs zero-loss trade-off.
• Future grid. Superconducting fault current limiters; potentially superconducting transmission lines for cities.

Common mistakes

• Treating superconducting as just "zero resistance". Has additional properties — Meissner effect, energy gap, quantization of flux.
• Confusing T_c with operating T. T_c is critical temperature. Real systems operate at significantly lower T to avoid fluctuations.
• Believing room-T SC has been achieved. Despite many claimed discoveries, no reproducible room-T ambient-pressure SC has been confirmed (as of 2024).
• Forgetting critical current/field limits. Above critical I or H, SC reverts to normal — and often dramatically (quench).
• Treating BCS as universal. BCS works for conventional SC; high-T_c cuprates and iron-based SC have different mechanisms (still being researched).
• Underestimating cooling complexity. Each T_c regime needs different cryogenic infrastructure — He (4 K) much harder than N₂ (77 K).