London equations

The London equations, developed by brothers Fritz and Heinz London in 1935, [cite journal
last= London
first= F.
coauthors= H. London
month= March
title= The Electromagnetic Equations of the Supraconductor
journal= Proc. Roy. Soc. (London)
volume= A149
url= http://www.jstor.org/sici?sici=0080-4630(19350301)149%3A866%3C71%3ATEEOTS%3E2.0.CO%3B2-2
year= 1935 ] 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. [cite book
author = Michael Tinkham
title = Introduction to Superconductivity
publisher = McGraw-Hill
year = 1996
id = ISBN 0-07-064878-6] [cite book
author = Neil W. Ashcroft
coauthors = N. David Mermin
title = Solid State Physics
publisher = Saunders College
year = 1976
pages = 738
id = ISBN 0-03-083993-9] [cite book
author = Charles Kittel
title = Introduction to Solid State Physics
publisher = Something
year = 1999
id = ISBN 0-03-083993-9] A major triumph of the equations is their ability to explain the Meissner effect, [cite journal
last= Meissner
first= W.
title=Ein neuer Effekt bei Eintritt der Supraleitfähigkeit
coauthors= R. Ochsenfeld
journal= Naturwissenschaften
volume= 21
year= 1933 ] wherein a material exponentially expels all internal magnetic fields as it crosses the superconducting threshold.

Formulations

There are two London equations when expressed in terms of measurable fields: :$frac\{partial\; mathbf\{j\}\_s\}\{partial\; t\}\; =\; frac\{n\_s\; e^2\}\{m\}mathbf\{E\},\; qquad\; mathbf\{\; abla\}\; imesmathbf\{j\}\_s\; =-frac\{n\_s\; e^2\}\{mc\}mathbf\{B\}.$ Here $\{mathbf\{j\_s$ is the superconducting current, E and B are respectively the electric and magnetic fields within the superconductor, $e,$ is the charge of an electron, $m,$ is electron mass, and $n\_s,$ is a phenomenological constant loosely associated with a number density of superconducting carriers [cite book
author = James F. Annett
title = Superconductivity, Superfluids and Condensates
publisher = Oxford
year = 2004
pages = 58
id = ISBN 0 19 850756 9] .Throughout this article Gaussian (cgs) units are employed.

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" [cite book
author = James F. Annett
title = Superconductivity, Superfluids and Condensates
publisher = Oxford
year = 2004
pages = 58
id = ISBN 0 19 850756 9] [cite book
author = John David Jackson
title = Classical Electrodynamics
publisher = John Wiley & Sons
year = 1999
pages = 604
id = ISBN 0 19 850756 9] in terms of the vector potential A::$mathbf\{j\}\_s\; =-frac\{n\_se\_s^2\}\{mc\}mathbf\{A\}.$

The last equation suffers from only the disadvantage that it is not gauge invariant, but is true only in the London gauge, where the divergence of A is zero. [cite book
author = Michael Tinkham
title = Introduction to Superconductivity
publisher = McGraw-Hill
year = 1996
pages = 6
id = ISBN 0-07-064878-6]

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".)] :$abla\; imes\; mathbf\{B\}\; =\; frac\{4\; pi\; mathbf\{j\{c\}$,then the result is the differential equation:$abla^2\; mathbf\{B\}\; =\; frac\{1\}\{lambda^2\}mathbf\{B\},\; qquad\; lambda\; equiv\; sqrt\{frac\{m\; c^2\}\{4\; pi\; n\_s\; e^2.$Thus, the London equations imply a characteristic length scale, $lambda$, over which external magnetic fields are exponentially suppressed. This value is the London penetration depth.

A simple example geometry is a flat boundary between 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

:$B\_z(x)\; =\; B\_0\; e^\{-x\; /\; lambda\}.\; ,$

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 derived in any conventional sense of the word, [cite book
author = Michael Tinkham
title = Introduction to Superconductivity
publisher = McGraw-Hill
year = 1996
pages = 5
id = ISBN 0-07-064878-6] 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 brothers London imagined electrons as if they were free electrons under the influence of a uniform external electric field. According to the Lorentz force law :$mathbf\{F\}=emathbf\{E\}+\; frac\{e\}\{c\}\; mathbf\{v\}\; imes\; mathbf\{B\}$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,:$abla\; imes\; mathbf\{E\}\; =\; -frac\{1\}\{c\}frac\{partial\; mathbf\{B\{partial\; t\}$,to obtain:$frac\{partial\}\{partial\; t\}left(\; abla\; imes\; mathbf\{j\}\_s\; +\; frac\{n\_s\; e^2\}\{m\; c\}\; mathbf\{B\}\; ight)\; =\; 0.$

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. [cite book
author = John David Jackson
title = Classical Electrodynamics
publisher = John Wiley & Sons
year = 1999
pages = 603-604
id = ISBN 0 19 850756 9] [cite book
author = Michael Tinkham
title = Introduction to Superconductivity
publisher = McGraw-Hill
year = 1996
pages = 5-6
id = ISBN 0-07-064878-6] Current density is defined according to the equation:$mathbf\{j\}\_s\; =\; n\_s\; e\; mathbf\{v\}.$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 is given by:$mathbf\{v\}\; =\; frac\{1\}\{m\}\; left(\; mathbf\{p\}\; -\; frac\{e\}\{c\}mathbf\{A\}\; ight).$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, [cite book
author = Michael Tinkham
title = Introduction to Superconductivity
publisher = McGraw-Hill
year = 1996
pages = 5
id = ISBN 0-07-064878-6] in such a state the canonical momentum p is zero. This leaves:$mathbf\{j\}\_s\; =-frac\{n\_se\_s^2\}\{mc\}mathbf\{A\},$ which is the London equation according to the second formulation above.

References