| London equations | |
|---|---|
| Name | London equations |
| Field | Condensed matter physics |
| Introduced | 1935 |
| Authors | Fritz London and Heinz London |
| Related | Superconductivity, Meissner effect, Ginzburg–Landau theory |
London equations
The London equations are a pair of phenomenological electrodynamic relations introduced by Fritz London and Heinz London in 1935 to describe the electromagnetic response of superconductors. They provide a simple macroscopic description of perfect conductivity and the Meissner effect, establishing a bridge between classical electrodynamics and microscopic theories of superconductivity such as BCS theory. Their importance in Quantum Physics lies in constraining field behavior in coherent quantum states and guiding later theoretical developments.
The London equations model the current density in a superconducting condensate in response to electric and magnetic fields, capturing key quantum-coherent phenomena at macroscopic scales. By introducing a rigidity of the superconducting phase and a characteristic penetration depth, the Londons supplied testable predictions about magnetic field exclusion that informed experiments at institutions like the University of Cambridge and laboratories including Bell Labs and Cavendish Laboratory. In the wider context of quantum theory, they exemplify how collective quantum effects produce novel emergent electrodynamics, influencing later work by Lev Landau, Vitaly Ginzburg, and John Bardeen.
The equations were formulated by the brothers Fritz and Heinz London while Fritz was at University College London and later at University of Oxford and Kaiser Wilhelm Institute associates. Their work followed the discovery of superconductivity by Heike Kamerlingh Onnes in 1911 and experimental studies of the Meissner effect by Walther Meissner and Robert Ochsenfeld in 1933. The Londons proposed their relations in the mid-1930s to resolve discrepancies between perfect conductivity models and observed magnetic field expulsion; their papers influenced subsequent theoretical efforts by Lev Landau and the phenomenological Ginzburg–Landau theory paper of 1950, and ultimately fed into microscopic explanations by Bardeen–Cooper–Schrieffer in 1957.
The London equations are usually presented as two coupled relations for the superconducting current density J_s and the electromagnetic fields E and B. In differential form they read: - First London equation: dJ_s/dt = (n_s e^2/m) E, expressing acceleration of the supercurrent charge carriers of density n_s, charge e and effective mass m. - Second London equation: ∇ × J_s = −(n_s e^2/m) B, relating current vorticity to magnetic flux density and implying an exponential falloff of B inside a superconductor with characteristic penetration depth λ_L = sqrt(m/(μ_0 n_s e^2)). These relations combine with Maxwell's equations to yield the Helmholtz equation for B inside a superconductor and predict zero resistivity and frozen-in phase behavior. The equations are often written using the superconducting carrier velocity and canonical momentum, connecting to the notion of a macroscopic wavefunction introduced later in Ginzburg–Landau theory and used in modern treatments with gauge fields.
The second London equation directly explains the Meissner effect: applying ∇× to Maxwell–Ampère's law and using the London relation yields an exponential screening of magnetic fields, with B(z) ∝ exp(−z/λ_L) from the surface. This distinguishes superconductors from ideal classical conductors, which would trap magnetic flux. The Londons thus provided a compact account of magnetic flux exclusion and introduced λ_L as a material parameter measured in metals like lead and niobium and in alloy and compound superconductors. Their framework also clarifies flux quantization in multiply connected geometries when combined with quantum phase considerations, an effect later observed in SQUID devices and experiments by groups at Stanford University and MIT.
While originally phenomenological, the London equations can be motivated from a quantum description of a macroscopic coherent state. Assuming a superconducting condensate described by a single-valued macroscopic wavefunction ψ = √n_s e^{iφ}, the supercurrent J_s ∝ n_s (ħ/m) (∇φ − (e/ħ) A) leads, under the London gauge choice ∇·A = 0 and constant n_s, to the second London equation. This connects the equations to canonical quantization and gauge invariance in quantum mechanics and quantum electrodynamics, and shows how phase rigidity of the condensate underlies electromagnetic rigidity. The gauge choice often used in derivations is named the London gauge, convenient for eliminating scalar potentials in static problems and emphasizing the role of the vector potential A and phase φ.
Experiments throughout the 20th century confirmed London predictions: measurements of penetration depths using microwave resonators, muon spin rotation at facilities such as TRIUMF and ISIS Neutron and Muon Source, and magnetization studies in materials characterized at Argonne National Laboratory and Oak Ridge National Laboratory. The London model informs the design and interpretation of superconducting magnets, particle accelerator cavities at CERN and Fermilab, and superconducting electronics such as Josephson junction circuits. Screening lengths measured in conventional superconductors generally match London estimates when corrected for temperature dependence via two-fluid models and later microscopic results.
The London equations are limited by their assumptions of constant n_s and neglect of coherence effects near critical temperature T_c and vortex cores. They do not predict the critical field H_c, the coherence length ξ, or the detailed temperature dependence of order parameters; these are addressed by Ginzburg–Landau theory and microscopic BCS theory. Ginzburg–Landau introduces a complex order parameter with spatial variation and free-energy functionals, yielding both λ and ξ and predicting type-I and type-II behavior and vortex lattices discovered by Abrikosov. BCS provides a microscopic derivation of n_s(T) and links to quasiparticle excitations; combined with London concepts it clarifies limits of validity and corrections, including nonlocal electrodynamics by Pippard and two-fluid models by Gorter–Casimir.
Category:Superconductivity Category:Condensed matter physics