London equations

London equations. The London equations are a pair of phenomenological partial differential equations formulated by the brothers Fritz London and Heinz London in 1935 to describe the electromagnetic properties of superconductors. They provided the first successful theoretical explanation of the Meissner effect, the complete expulsion of magnetic fields from a superconductor's interior. This work established a foundational framework for understanding electrodynamics in the superconducting state, predating the microscopic BCS theory by over two decades.

Introduction and historical context

The development of the London equations followed the landmark discovery of superconductivity by Heike Kamerlingh Onnes in Leiden in 1911. For years, the zero electrical resistance of materials like mercury and lead at low temperatures remained a profound mystery. A pivotal breakthrough came in 1933 with the discovery of the Meissner effect by Walther Meissner and Robert Ochsenfeld at the Physikalisch-Technische Bundesanstalt in Berlin, demonstrating that superconductors were not just perfect conductors but also perfect diamagnets. This finding challenged existing theories and prompted the London brothers, who were working at the University of Oxford after fleeing Nazi Germany, to seek a new electromagnetic description. Their work was influenced by contemporary ideas in quantum mechanics, particularly the concept of a macroscopic wave function, and built upon earlier attempts by Cornelius Gorter and Hendrik Casimir.

Formulation and mathematical description

The London equations consist of two key relations connecting the supercurrent density \(\mathbf{J}_s\) to the electromagnetic fields. The first London equation describes the dynamics of the supercurrent in response to an electric field \(\mathbf{E}\): \(\frac{\partial \mathbf{J}_s}{\partial t} = \frac{n_s e^2}{m} \mathbf{E}\), where \(n_s\) is the density of superconducting electrons, \(e\) is the elementary charge, and \(m\) is the electron mass. This implies an acceleration of superconducting carriers without scattering, leading to zero resistance. The second London equation, more crucial for explaining the Meissner effect, relates the supercurrent to the magnetic field \(\mathbf{B}\): \(\nabla \times \mathbf{J}_s = -\frac{n_s e^2}{m} \mathbf{B}\). When combined with Ampère's law from Maxwell's equations, \(\nabla \times \mathbf{B} = \mu_0 \mathbf{J}_s\), it yields the equation \(\nabla^2 \mathbf{B} = \frac{\mathbf{B}}{\lambda_L^2}\), where \(\lambda_L = \sqrt{\frac{m}{\mu_0 n_s e^2}}\) is the London penetration depth. This predicts an exponential decay of magnetic fields inside a superconductor over this characteristic length.

Physical interpretation and significance

Physically, the London equations postulate that a superconductor contains a frictionless, quantum-mechanical condensate of charged particles. The second equation implies that the canonical momentum of these superconducting electrons is zero in the ground state, a condition now understood as the rigidity of the macroscopic wave function phase. The primary triumph was providing a quantitative explanation for the Meissner effect, showing that magnetic flux is expelled except in a thin surface layer of thickness \(\lambda_L\). This established superconductivity as a distinct thermodynamic phase and introduced the critical concept of penetration depth, a fundamental parameter measurable in experiments. The equations also successfully described other phenomena, such as the dependence of the critical field on sample geometry and aspects of the Little–Parks effect.

Limitations and relation to BCS theory

While groundbreaking, the London theory is a local, phenomenological model with several limitations. It assumes a constant superconducting electron density \(n_s\) and does not account for its spatial variation near surfaces or in magnetic fields, a shortcoming addressed by the later Ginzburg–Landau theory developed by Vitaly Ginzburg and Lev Landau. More fundamentally, it does not explain the microscopic origin of superconductivity. This was ultimately provided by the BCS theory of John Bardeen, Leon Cooper, and John Robert Schrieffer in 1957. BCS theory identifies the superconducting carriers as Cooper pairs of electrons bound by lattice interactions, justifying the London equations from first principles. In the BCS framework, the London penetration depth and superfluid density are temperature-dependent quantities, a refinement beyond the original simple model.

Applications and experimental verification

The London equations remain essential for analyzing the electromagnetic response of conventional type-I superconductors. They are routinely used to interpret measurements of penetration depth using techniques like muon spin rotation, and are fundamental to the design of superconducting devices such as SQUID magnetometers and RF cavities in particle accelerators like the Large Hadron Collider. Experimental verification of the Meissner effect and the exponential field decay is a standard demonstration in low-temperature physics labs worldwide. The concepts directly underpin the operation of maglev trains and are critical for understanding vortex states in type-II superconductors, although the latter requires extensions like the Ginzburg–Landau theory. The equations' prediction of flux quantization, later confirmed experimentally, was a key step toward the development of Ginzburg–Landau theory and the complete theoretical picture of superconductivity. Category:Superconductivity Category:Equations Category:Condensed matter physics