London equations

 ( Superconductivity ) 

The London equations, developed by brothers Fritz London and Heinz London in 1935, are constitutive relations for a superconductor relating its superconducting current to electromagnetic fields in and around it. Whereas Ohm's law is the simplest constitutive relation for an ordinary conductor, the London equations are the simplest meaningful description of superconducting phenomena, and form the genesis of almost any modern introductory text on the subject.

Neil Ashcroft (1976). 9780030839931, Saunders College. . ISBN 9780030839931

Charles Kittel (2025). 047141526X, Wiley. 047141526X

A major triumph of the equations is their ability to explain the Meissner effect, wherein a material exponentially expels all internal magnetic fields as it crosses the superconducting threshold.

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Description There are two London equations when expressed in terms of measurable fields:

$\backslash frac\{\backslash partial\; \backslash mathbf\{j\}\_\{\backslash rm\; s\}\}\{\backslash partial\; t\}\; =\; \backslash frac\{n\_\{\backslash rm\; s\}\; e^2\}\{m\}\backslash mathbf\{E\},\; \backslash qquad\; \backslash mathbf\{\backslash nabla\}\backslash times\backslash mathbf\{j\}\_\{\backslash rm\; s\}\; =-\backslash frac\{n\_\{\backslash rm\; s\}\; e^2\}\{m\; \}\backslash mathbf\{B\}.$

Here $\{\backslash mathbf\{j\}\}\_\{\backslash rm\; s\}$ is the (superconducting) current density, E and B are respectively the electric and magnetic fields within the superconductor, $e\backslash ,$ is the charge of an electron or proton, $m\backslash ,$ is electron mass, and $n\_\{\backslash rm\; s\}\backslash ,$ is a phenomenological constant loosely associated with a number density of superconducting carriers.

James F. Annett (2025). 9780198507567, Oxford. . ISBN 9780198507567

The two equations can be combined into a single "London Equation"

in terms of a specific vector potential $\backslash mathbf\{A\}\_\{\backslash rm\; s\}$ which has been Gauge fixing to the "London gauge", giving:

$\backslash mathbf\{j\}\_\{s\}\; =-\backslash frac\{n\_\{\backslash rm\; s\}e^2\}\{m\}\backslash mathbf\{A\}\_\{\backslash rm\; s\}.$

In the London gauge, the vector potential obeys the following requirements, ensuring that it can be interpreted as a current density:

Michael Tinkham (1996). 9780070648784, McGraw-Hill. . ISBN 9780070648784

• $\backslash nabla\backslash cdot\; \backslash mathbf\{A\}\_\{\backslash rm\; s\}\; =\; 0,$
• $\backslash mathbf\{A\}\_\{\backslash rm\; s\}\; =\; 0$ in the superconductor bulk,
• $\backslash mathbf\{A\}\_\{\backslash rm\; s\}\; \backslash cdot\; \backslash hat\{\backslash mathbf\{n\}\}\; =\; 0,$ where $\backslash hat\{\backslash mathbf\{n\}\}$ is the normal vector at the surface of the superconductor.

The first requirement, also known as Gauge fixing condition, leads to the constant superconducting electron density $\backslash dot\; \backslash rho\_\{\backslash rm\; s\}\; =\; 0$ as expected from the continuity equation. The second requirement is consistent with the fact that supercurrent flows near the surface. The third requirement ensures no accumulation of superconducting electrons on the surface. These requirements do away with all gauge freedom and uniquely determine the vector potential. One can also write the London equation in terms of an arbitrary gauge $\backslash mathbf\{A\}$ by simply defining $\backslash mathbf\{A\}\_\{\backslash rm\; s\}\; =\; (\backslash mathbf\{A\}\; +\; \backslash nabla\; \backslash phi)$, where $\backslash phi$ is a scalar function and $\backslash nabla\; \backslash phi$ is the change in gauge which shifts the arbitrary gauge to the London gauge. The vector potential expression holds for magnetic fields that vary slowly in space.

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London penetration depth If the second of London's equations is manipulated by applying Ampere's law,(The displacement is ignored because it is assumed that electric field only varies slowly with respect to time, and the term is already suppressed by a factor of c.)

$\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash mu\_0\; \backslash mathbf\{j\}$,

then it can be turned into the Helmholtz equation for magnetic field:

$\backslash nabla^2\; \backslash mathbf\{B\}\; =\; \backslash frac\{1\}\{\backslash lambda\_\{\backslash rm\; s\}^2\}\backslash mathbf\{B\}$

where the inverse of the laplacian eigenvalue:

$\backslash lambda\_\{\backslash rm\; s\}\; \backslash equiv\; \backslash sqrt\{\backslash frac\{m\}\{\backslash mu\_0\; n\_\{\backslash rm\; s\}\; e^2\}\}$

is the characteristic length scale, $\backslash lambda\_\{\backslash rm\; s\}$, over which external magnetic fields are exponentially suppressed: it is called the London penetration depth: typical values are from 50 to 500 nanometers.

For example, consider a superconductor within free space where the magnetic field outside the superconductor is a constant value pointed parallel to the superconducting boundary plane in the z direction. If x leads perpendicular to the boundary then the solution inside the superconductor may be shown to be

$B\_z(x)\; =\; B\_0\; e^\{-x\; /\; \backslash lambda\_\{\backslash rm\; s\}\}.\; \backslash ,$

From here the physical meaning of the London penetration depth can perhaps most easily be discerned.

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Rationale

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Original arguments While it is important to note that the above equations cannot be formally derived,

Michael Tinkham (1996). 9780070648784, McGraw-Hill. . ISBN 9780070648784

the Londons did follow a certain intuitive logic in the formulation of their theory. Substances across a stunningly wide range of composition behave roughly according to Ohm's law, which states that current is proportional to electric field. However, such a linear relationship is impossible in a superconductor for, almost by definition, the electrons in a superconductor flow with no resistance whatsoever. To this end, the London brothers imagined electrons as if they were free electrons under the influence of a uniform external electric field. According to the Lorentz force law

$\backslash mathbf\{F\}=m\; \backslash dot\{\backslash mathbf\{v\}\}=-e\backslash mathbf\{E\}\; -\; e\backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\}$

these electrons should encounter a uniform force, and thus they should in fact accelerate uniformly. Assume that the electrons in the superconductor are now driven by an electric field, then according to the definition of current density $\backslash mathbf\{j\}\_\{\backslash rm\; s\}\; =\; -n\_\{\backslash rm\; s\}\; e\; \backslash mathbf\{v\}\_\{\backslash rm\; s\}$we should have

$\backslash frac\{\backslash partial\; \backslash mathbf\{j\}\_\{s\}\}\{\backslash partial\; t\}\; =\; -n\_\{\backslash rm\; s\}\; e\; \backslash frac\{\backslash partial\; \backslash mathbf\{v\}\}\{\backslash partial\; t\; \}\; =\backslash frac\{n\_\{\backslash rm\; s\}\; e^2\}\{m\}\backslash mathbf\{E\}$

This is the first London equation. To obtain the second equation, take the curl of the first London equation and apply Faraday's law,

$\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; -\backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\{\backslash partial\; t\}$,

to obtain

$\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\backslash left(\; \backslash nabla\; \backslash times\; \backslash mathbf\{j\}\_\{\backslash rm\; s\}\; +\; \backslash frac\{n\_\{\backslash rm\; s\}\; e^2\}\{m\}\; \backslash mathbf\{B\}\; \backslash right)\; =\; 0.$

As it currently stands, this equation permits both constant and exponentially decaying solutions. The Londons recognized from the Meissner effect that constant nonzero solutions were nonphysical, and thus postulated that not only was the time derivative of the above expression equal to zero, but also that the expression in the parentheses must be identically zero:

$\backslash nabla\; \backslash times\; \backslash mathbf\{j\}\_\{\backslash rm\; s\}\; +\; \backslash frac\{n\_\{\backslash rm\; s\}\; e^2\}\{m\}\; \backslash mathbf\{B\}\; =\; 0$

This results in the second London equation and $\backslash mathbf\{j\}\_\{s\}\; =-\backslash frac\{n\_\{\backslash rm\; s\}e^2\}\{m\}\backslash mathbf\{A\}\_\{\backslash rm\; s\}$ (up to a gauge transformation which is fixed by choosing "London gauge") since the magnetic field is defined through $B=\backslash nabla\; \backslash times\; A\_\{\backslash rm\; s\}.$

Additionally, according to Ampere's law $\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash mu\_0\; \backslash mathbf\{j\}\_\{\backslash rm\; s\}$ , one may derive that: $\backslash nabla\; \backslash times\; (\backslash nabla\backslash times\; \backslash mathbf\{B\})\; =\backslash nabla\; \backslash times\; \backslash mu\_0\; \backslash mathbf\{j\}\_\{\backslash rm\; s\}\; =-\backslash frac\{\backslash mu\_0\; n\_\{\backslash rm\; s\}\; e^2\}\{m\}\; \backslash mathbf\{B\}.$

On the other hand, since $\backslash nabla\; \backslash cdot\; \backslash mathbf\{B\}\; =\; 0$, we have $\backslash nabla\; \backslash times\; (\backslash nabla\backslash times\; \backslash mathbf\{B\})\; =\; -\backslash nabla^2\backslash mathbf\{B\}$, which leads to the spatial distribution of magnetic field obeys :

$\backslash nabla^2\; \backslash mathbf\{B\}\; =\; \backslash frac\{1\}\{\backslash lambda\_\{\backslash rm\; s\}^2\}\backslash mathbf\{B\}$

with penetration depth $\backslash lambda\_\{\backslash rm\; s\}=\backslash sqrt\{\backslash frac\{m\}\{\backslash mu\_0\; n\_\{\backslash rm\; s\}\; e^2\}\}$. In one dimension, such Helmholtz equation has the solution form $B\_z(x)\; =\; B\_0\; e^\{-x\; /\; \backslash lambda\_\{\backslash rm\; s\}\}.\; \backslash ,$

Inside the superconductor $(x>0)$, the magnetic field exponetially decay, which well explains the Meissner effect. With the magnetic field distribution, we can use Ampere's law $\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash mu\_0\; \backslash mathbf\{j\}\_\{\backslash rm\; s\}$ again to see that the supercurrent $\backslash mathbf\{j\}\_\{\backslash rm\; s\}$also flows near the surface of superconductor, as expected from the requirement for interpreting $\backslash mathbf\{j\}\_\{\backslash rm\; s\}$as physical current.

While the above rationale holds for superconductor, one may also argue in the same way for a perfect conductor. However, one important fact that distinguishes the superconductor from perfect conductor is that perfect conductor does not exhibit Meissner effect for

$\backslash nabla^2\; \backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\{\backslash partial\; t\}\; =\; \backslash frac\{1\}\{\backslash lambda\_\{\backslash rm\; s\}^2\}\backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\{\backslash partial\; t\}.$

For

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Canonical momentum arguments It is also possible to justify the London equations by other means.

John David Jackson (1999). 9780198507567, John Wiley & Sons. . ISBN 9780198507567

Michael Tinkham (1996). 9780070648784, McGraw-Hill. . ISBN 9780070648784

Current density is defined according to the equation

$\backslash mathbf\{j\}\_\{\backslash rm\; s\}\; =-\; n\_\{\backslash rm\; s\}\; e\; \backslash mathbf\{v\}\_\{\backslash rm\; s\}.$

Taking this expression from a classical description to a quantum mechanical one, we must replace values $\backslash mathbf\{j\}\_\{\backslash rm\; s\}$ and $\backslash mathbf\{v\}\_\{\backslash rm\; s\}$ by the expectation values of their operators. The velocity operator

$\backslash mathbf\{v\}\_\{\backslash rm\; s\}\; =\; \backslash frac\{1\}\{m\}\; \backslash left(\; \backslash mathbf\{p\}\; +\; e\; \backslash mathbf\{A\}\_\{\backslash rm\; s\}\; \backslash right)$

is defined by dividing the gauge-invariant, kinematic momentum operator by the particle mass m.

L. D. Landau and E. M. Lifshitz (1977). 9780750635394, Butterworth-Heinemann. ISBN 9780750635394

Note we are using $-e$ as the electron charge. We may then make this replacement in the equation above. However, an important assumption from the Bcs theory is that the superconducting state of a system is the ground state, and according to a theorem of Bloch's,Tinkham p.5: "This theorem is apparently unpublished, though famous." in such a state the canonical momentum p is zero. This leaves

$\backslash mathbf\{j\}\; =-\backslash frac\{n\_\{\backslash rm\; s\}e^2\}\{m\}\backslash mathbf\{A\}\_\{\backslash rm\; s\},$

which is the London equation according to the second formulation above.

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