London equations

London equations
Name	London equations
Caption	Phenomenological description of superconducting electrodynamics
Field	Condensed matter physics
Introduced	1935
Authors	Fritz London; Heinz London
Related	Meissner effect; superconductivity; BCS theory

London equations The London equations are a pair of coupled relations describing the electrodynamics of superconductors within a phenomenological framework. They provide a concise macroscopic account of zero electrical resistance and the expulsion of magnetic flux in bulk superconducting materials, linking superconducting current density to electromagnetic fields. These equations served as a bridge between early experimental discoveries of superconductivity and later microscopic theories, influencing developments in low-temperature physics and materials science.

Introduction

The London equations were formulated to capture two hallmark phenomena observed in superconducting metals and alloys during the early 20th century: persistent currents and magnetic-field exclusion. Proposed by Fritz London and Heinz London, the relations treat the superconducting condensate as a charged fluid characterized by a density and an effective mass, yielding simple differential equations connecting the superconducting current density to electric and magnetic fields. Their phenomenological nature made them directly applicable to experiments on elemental superconductors such as lead and mercury, and they informed interpretations of results from low-temperature research at institutions like the Kamerlingh Onnes laboratory and later work at the Cavendish Laboratory.

Derivation and mathematical formulation

Starting from the assumption of a dissipationless superconducting carrier density ns with charge e and mass m, the first London relation expresses the time derivative of the superconducting current density Js as proportional to the electric field E: m ∂Js/∂t = nse^2 E. The second London relation asserts that the curl of the superconducting current density is proportional to the magnetic induction B: ∇ × Js = - (nse^2 / m) B. Combining these with Maxwell's equations yields a modified Helmholtz-type equation for B: ∇^2 B = λ_L^{-2} B, where λ_L = sqrt(m / (μ0 nse^2)) defines the London penetration depth. In static situations the relations predict exponential decay of magnetic fields into a superconducting half-space; in dynamic contexts they give rise to nondissipative charge acceleration and plasma-like collective modes. The formulation assumes uniform ns and neglects explicit pairing, while preserving gauge invariance through minimal coupling to the vector potential A in the canonical momentum p = m v + e A.

Physical implications and Meissner effect

The second London relation directly yields the Meissner effect: an applied magnetic field is expelled from the superconducting bulk, penetrating only to the characteristic length λ_L. This contrasts with perfect conductors that would merely freeze-in preexisting magnetic flux; the London's relations predict active expulsion when a material transitions into the superconducting state. Consequences include quantifiable magnetization curves for Type I materials like tin and aluminium, the formation of screening currents near surfaces, and conditions for magnetic field trapping in Type II materials such as niobium and vanadium when vortices are present. The penetration depth λ_L depends on carrier density and effective mass, linking measurable magnetic response to microscopic electronic properties investigated in experiments at facilities like the Bell Labs and the Leiden cryogenics groups.

Applications and experimental tests

Experimental determinations of the London penetration depth via techniques such as muon spin rotation, microwave surface impedance, and AC susceptibility have tested the quantitative predictions of the London framework in conventional superconductors. The equations underpin the design and interpretation of superconducting magnet systems used in particle accelerator laboratories like CERN and in medical imaging with superconducting magnets for magnetic resonance imaging developed by clinical research teams. They inform modeling of superconducting resonators and Josephson junction arrays used in quantum computing research at institutions including IBM and various university laboratories. Precision measurements of magnetic screening, flux quantization observed in superconducting rings, and penetration-depth temperature dependence in elemental superconductors have validated and constrained the phenomenological parameters appearing in the London relations.

Limitations and extensions

Although successful phenomenologically, the London relations omit microscopic mechanisms such as Cooper pairing and energy gap formation addressed by the Bardeen–Cooper–Schrieffer formalism. They assume a rigid, single-component superfluid density and cannot capture spatial variations on the scale of the superconducting coherence length ξ, vortex core structure, or strong-coupling effects. Extensions include the Ginzburg–Landau theory, which introduces an order parameter field and reproduces London behavior in appropriate limits, and microscopic derivations from BCS theory that yield temperature-dependent penetration depths and nonlocal electrodynamics (Pippard kernel). Further generalizations encompass anisotropic and multiband superconductors, treated in advanced models applied to materials discovered in the late 20th and early 21st centuries, including high-temperature cuprate superconductors and iron-based compounds studied at laboratories such as Brookhaven and the Max Planck Institute.

Historical context and contributors

The London relations were introduced in the 1930s by brothers Fritz London and Heinz London following early experimental advances by Heike Kamerlingh Onnes, whose discovery of superconductivity catalyzed theoretical attempts to explain zero resistance and magnetic behavior. Subsequent theoretical developments involved Lev Landau and Vitaly Ginzburg, whose Ginzburg–Landau theory extended phenomenological descriptions, and later microscopic resolution by John Bardeen, Leon Cooper, and Robert Schrieffer in the BCS theory. Experimentalists including Walther Meissner and Robert Ochsenfeld provided pivotal observations of magnetic flux expulsion, while later contributors such as Brian Josephson, Alexei Abrikosov, and Philip Anderson expanded the field to include weak links, vortex matter, and disorder effects. The interplay between theory and experiment at institutions like the Royal Society, the National Bureau of Standards, and university laboratories worldwide shaped the modern understanding of superconductivity.

Category:Superconductivity