| London equations | |
|---|---|
| Name | London equations |
| Caption | Phenomenological description of superconducting electrodynamics |
| Discovered | 1935 |
| Discoverer | Fritz London; Heinz London |
| Field | Condensed matter physics; Superconductivity |
| Related | Meissner effect; Ginzburg–Landau theory; BCS theory; Maxwell's equations |
London equations
The London equations are a pair of phenomenological relations introduced by Fritz London and Heinz London in 1935 to describe electrodynamics inside superconductors. They capture the key features of perfect conductivity and the Meissner effect, providing a macroscopic bridge between classical electrodynamics and quantum descriptions of superconductivity such as BCS theory. The equations remain central to understanding penetration depth, magnetic response, and collective modes in superconducting systems.
The London equations were proposed shortly after the discovery of superconductivity and the experimental identification of the Meissner effect by Walther Meissner and Robert Ochsenfeld (1933). Fritz and Heinz London formulated a simple phenomenological model that supplemented Maxwell's equations with constitutive relations for the superconducting current. Their work predates microscopic theories like the BCS theory (1957) by John Bardeen, Leon Cooper, and Robert Schrieffer and influenced subsequent macroscopic approaches such as the Ginzburg–Landau theory (1950) developed by Vitaly Ginzburg and Lev Landau. The Londons emphasized a quantum coherent condensate interpretation, anticipating concepts like off-diagonal long-range order used later by Philip W. Anderson and others.
The London equations consist of two coupled relations for the superconducting current density J_s and the electromagnetic fields E and B:
- First London equation (temporal response): ∂J_s/∂t = (n_s e^2 / m) E
- Second London equation (spatial response): ∇ × J_s = - (n_s e^2 / m) B
Here n_s is the density of superconducting carriers, e their charge, and m an effective mass. Combining the second London equation with Maxwell's curl equation ∇ × B = μ_0 J_total yields the exponential screening of magnetic fields inside a superconductor with the characteristic London penetration depth λ_L = sqrt(m / (μ_0 n_s e^2)). The equations are often written in gauge-invariant form using the superconducting phase and vector potential A, linking to the Josephson relations of Brian D. Josephson and the concept of flux quantization in superconducting rings first observed in experiments at institutions such as Rutherford Appleton Laboratory and Bell Labs.
Physically, the first London equation expresses an ideal inertia-free acceleration of the superconducting carriers by an electric field, accounting for zero DC resistivity. The second equation enforces a proportionality between current circulation and magnetic field that eliminates interior magnetization, explaining the Meissner effect rather than mere perfect conductivity. The role of n_s connects the equations to temperature dependence and the two-fluid model by Fritz Bloch and others, where a normal component coexists with a superfluid condensate. In superconductors with broken time-reversal or unconventional pairing (e.g., high-temperature superconductors), corrections to London behavior can appear; nevertheless, London relations provide first-order estimates for observables like λ_L and the superconducting plasma frequency measured by techniques at CERN-linked collaborations and national laboratories.
Phenomenologically, the Londons motivated their relations by combining gauge invariance with the notion of a rigid macroscopic wavefunction for the superconducting condensate. In microscopic terms, the London equations emerge from the zero-temperature, long-wavelength limit of BCS theory and from linear response calculations using the Gor'kov formalism developed by Lev Gor'kov. Within BCS, the supercurrent is carried by Cooper pairs with charge 2e and an effective mass 2m, leading to expressions for n_s that depend on the superconducting gap Δ and the quasiparticle spectrum. Derivations using the path integral or Green's function methods connect London behavior to Anderson's Higgs mechanism in condensed matter, where the Anderson–Higgs mechanism gives mass to the photon inside a superconductor, analogous to symmetry-breaking concepts in particle physics.
London equations underpin analysis of many experiments: measurements of the penetration depth λ_L via muon spin rotation (μSR) at facilities like TRIUMF and Paul Scherrer Institute, microwave resonator studies at MIT and Stanford University, and magnetic imaging with scanning SQUIDs developed at University of Twente and IBM Research. They are used to model screening in superconducting cavities for particle accelerators at CERN and SLAC National Accelerator Laboratory, flux trapping in superconducting qubits at Google and IBM, and interpreting optical conductivity in terahertz spectroscopy. Precision tests compare temperature and doping dependence of λ_L against predictions from BCS and alternative pairing theories in materials such as elemental lead (Pb), niobium (Nb), and cuprate high-Tc compounds studied at Bell Labs and university research groups.
While successful, the London equations are limited to low-frequency, long-wavelength, and linear-response regimes. They neglect vortex dynamics relevant to type-II superconductors described by Abrikosov vortex theory and viscous flux-flow resistivity studied by A. A. Abrikosov. Extensions include time-dependent Ginzburg–Landau theory for near-Tc dynamics, two-fluid and two-band London models for multiband superconductors like MgB2, and nonlocal electrodynamics (Pippard kernel) addressing finite mean free path effects introduced by A. B. Pippard. Modern developments embed London-like relations into topological superconductivity, unconventional pairing symmetries, and hybrid platforms for quantum computing, where deviations from ideal London behavior inform decoherence and device engineering. Ongoing work at universities and national labs continues to refine microscopic inputs (n_s, effective mass, pairing symmetry) that parameterize London phenomenology for new materials and quantum technologies.
Category:Superconductivity Category:Condensed matter physics