# London equations

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The London equations, developed by brothers Fritz and Heinz London in 1935, [1] relate current to electromagnetic fields in and around a superconductor. Arguably the simplest meaningful description of superconducting phenomena, they form the genesis of almost any modern introductory text on the subject. [2] [3] [4] A major triumph of the equations is their ability to explain the Meissner effect, [5] wherein a material exponentially expels all internal magnetic fields as it crosses the superconducting threshold.

Fritz Wolfgang London was a German physicist and professor at Duke University. His fundamental contributions to the theories of chemical bonding and of intermolecular forces are today considered classic and are discussed in standard textbooks of physical chemistry. With his brother Heinz London, he made a significant contribution to understanding electromagnetic properties of superconductors with the London equations and was nominated for the Nobel Prize in Chemistry on five separate occasions.

Heinz London was a German-British physicist. Together with his brother Fritz London he was a pioneer in the field of superconductivity.

The Meissner effect is the expulsion of a magnetic field from a superconductor during its transition to the superconducting state. The German physicists Walther Meissner and Robert Ochsenfeld discovered this phenomenon in 1933 by measuring the magnetic field distribution outside superconducting tin and lead samples. The samples, in the presence of an applied magnetic field, were cooled below their superconducting transition temperature, whereupon the samples cancelled nearly all interior magnetic fields. They detected this effect only indirectly because the magnetic flux is conserved by a superconductor: when the interior field decreases, the exterior field increases. The experiment demonstrated for the first time that superconductors were more than just perfect conductors and provided a uniquely defining property of the superconductor state. The ability for the expulsion effect is determined by the nature of equilibrium formed by the neutralization within the unit cell of a superconductor.

## Formulations

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

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

Here ${\displaystyle {\mathbf {j} }_{s}}$ is the superconducting current density, E and B are respectively the electric and magnetic fields within the superconductor, ${\displaystyle e\,}$ is the charge of an electron & proton, ${\displaystyle m\,}$ is electron mass, and ${\displaystyle n_{s}\,}$ is a phenomenological constant loosely associated with a number density of superconducting carriers. [6] Throughout this article SI units are employed.

In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional area at a given point in space, its direction being that of the motion of the charges at this point. In SI base units, the electric current density is measured in amperes per square metre.

On the other hand, if one is willing to abstract away slightly, both the expressions above can more neatly be written in terms of a single "London Equation" [6] [7] in terms of the vector potential A:

In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose gradient is a given vector field.

${\displaystyle \mathbf {j} _{s}=-{\frac {n_{s}e^{2}}{m}}\mathbf {A} .}$

The last equation suffers from only the disadvantage that it is not gauge invariant, but is true only in the Coulomb gauge, where the divergence of A is zero. [8] This equation holds for magnetic fields that vary slowly in space. [4]

## London penetration depth

If the second of London's equations is manipulated by applying Ampere's law, [9]

${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {j} }$,

then the result is the differential equation

${\displaystyle \nabla ^{2}\mathbf {B} ={\frac {1}{\lambda ^{2}}}\mathbf {B} ,\qquad \lambda \equiv {\sqrt {\frac {m}{\mu _{0}n_{s}e^{2}}}}.}$

Thus, the London equations imply a characteristic length scale, ${\displaystyle \lambda }$, over which external magnetic fields are exponentially suppressed. This value is the London penetration depth.

In superconductors, the London penetration depth characterizes the distance to which a magnetic field penetrates into a superconductor and becomes equal to −1 times that of the magnetic field at the surface of the superconductor. Typical values of λL range from 50 to 500 nm.

For an 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

${\displaystyle B_{z}(x)=B_{0}e^{-x/\lambda }.\,}$

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

## Rationale for the London equations

### Original arguments

While it is important to note that the above equations cannot be formally derived, [10] 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

${\displaystyle \mathbf {F} =e\mathbf {E} +e\mathbf {v} \times \mathbf {B} }$

these electrons should encounter a uniform force, and thus they should in fact accelerate uniformly. This is precisely what the first London equation states.

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

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

to obtain

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

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. This results in the second London equation.

### Canonical momentum arguments

It is also possible to justify the London equations by other means. [11] [12] Current density is defined according to the equation

${\displaystyle \mathbf {j} _{s}=-n_{s}e\mathbf {v} .}$

Taking this expression from a classical description to a quantum mechanical one, we must replace values j and v by the expectation values of their operators. The velocity operator

${\displaystyle \mathbf {v} ={\frac {1}{m}}\left(\mathbf {p} +e\mathbf {A} \right)}$

is defined by dividing the gauge-invariant, kinematic momentum operator by the particle mass m. [13] Note we are using ${\displaystyle -e}$ as the electron charge. We may then make this replacement in the equation above. However, an important assumption from the microscopic theory of superconductivity is that the superconducting state of a system is the ground state, and according to a theorem of Bloch's, [10] in such a state the canonical momentum p is zero. This leaves

${\displaystyle \mathbf {j} _{s}=-{\frac {n_{s}e^{2}}{m}}\mathbf {A} ,}$

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

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## References

1. London, F.; London, H. (1935). "The Electromagnetic Equations of the Supraconductor". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 149 (866): 71. Bibcode:1935RSPSA.149...71L. doi:10.1098/rspa.1935.0048.
2. Michael Tinkham (1996). Introduction to Superconductivity. McGraw-Hill. ISBN   0-07-064878-6.
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4. Charles Kittel (2005). Introduction to Solid State Physics (8th ed.). Wiley. ISBN   0-471-41526-X.
5. Meissner, W.; R. Ochsenfeld (1933). "Ein neuer Effekt bei Eintritt der Supraleitfähigkeit". Naturwissenschaften. 21 (44): 787. Bibcode:1933NW.....21..787M. doi:10.1007/BF01504252.
6. James F. Annett (2004). Superconductivity, Superfluids and Condensates. Oxford. p. 58. ISBN   0-19-850756-9.
7. John David Jackson (1999). Classical Electrodynamics. John Wiley & Sons. p. 604. ISBN   0-19-850756-9.
8. Michael Tinkham (1996). Introduction to Superconductivity. McGraw-Hill. p. 6. ISBN   0-07-064878-6.
9. (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.)
10. Michael Tinkham (1996). Introduction to Superconductivity. McGraw-Hill. p. 5. ISBN   0-07-064878-6.
11. John David Jackson (1999). Classical Electrodynamics. John Wiley & Sons. pp. 603–604. ISBN   0-19-850756-9.
12. Michael Tinkham (1996). Introduction to Superconductivity. McGraw-Hill. pp. 5–6. ISBN   0-07-064878-6.
13. L. D. Landau and E. M. Lifshitz (1977). Quantum Mechanics- Non-relativistic Theory. Butterworth-Heinemann. pp. 455–458. ISBN   0-7506-3539-8.