The London equations, developed by brothers
Fritz
Fritz originated as a German nickname for Friedrich, or Frederick (''Der Alte Fritz'', and ''Stary Fryc'' were common nicknames for King Frederick II of Prussia and Frederick III, German Emperor) as well as for similar names including Fridolin a ...
and
Heinz London
Heinz London (Bonn, Germany 7 November 1907 – 3 August 1970) was a German-British physicist. Together with his brother Fritz London he was a pioneer in the field of superconductivity.
Biography
London was born in Bonn in a liberal Jewish-Ge ...
in 1935, are
constitutive relation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
s for a
superconductor relating its superconducting current to
electromagnetic fields
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical co ...
in and around it. Whereas
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
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.
A major triumph of the equations is their ability to explain the
Meissner effect
The Meissner effect (or Meissner–Ochsenfeld effect) is the expulsion of a magnetic field from a superconductor during its transition to the superconducting state when it is cooled below the critical temperature. This expulsion will repel a ne ...
, wherein a material exponentially expels all internal magnetic fields as it crosses the superconducting threshold.
Description
There are two London equations when expressed in terms of measurable fields:
:
Here
is the (superconducting)
current density, E and B are respectively the electric and magnetic fields within the superconductor,
is the charge of an electron or proton,
is electron mass, and
is a phenomenological constant loosely associated with a
number density of superconducting carriers.
The two equations can be combined into a single "London Equation"
in terms of a specific
vector potential
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.
Formally, given a vector field v, a ''vecto ...
which has been
gauge fixed to the "London gauge", giving:
:
In the London gauge, the vector potential obeys the following requirements, ensuring that it can be interpreted as a current density:
*
*
in the superconductor bulk,
*
where
is the
normal vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
at the surface of the superconductor.
The first requirment, also known as
Coulomb gauge
In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct co ...
condition, leads to the constant superconducting electron density
as expected from the continuity equation. The second requirment is consistent with the fact that supercurrent flows near the surface. The third requirment 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
by simply defining
, where
is a scalar function and
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.
London penetration depth
If the second of London's equations is manipulated by applying
Ampere's law,
:
,
then it can be turned into the
Helmholtz equation for magnetic field:
:
where the inverse of the
laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
eigenvalue:
:
is the characteristic length scale,
, over which external magnetic fields are exponentially suppressed: it is called the
London penetration depth
In superconductors, the London penetration depth (usually denoted as \lambda or \lambda_L) characterizes the distance to which a magnetic field penetrates into a superconductor and becomes equal to e^ times that of the magnetic field at the surface ...
: typical values are from 50 to 500
nm.
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
:
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,
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
Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
, 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
Lorentz is a name derived from the Roman surname, Laurentius, which means "from Laurentum". It is the German form of Laurence. Notable people with the name include:
Given name
* Lorentz Aspen (born 1978), Norwegian heavy metal pianist and keyboar ...
:
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 dentisy
we should have
This is the first London equation. To obtain the second equation, take the curl of the first London equation and apply
Faraday's law,
:
,
to obtain
:
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 and
(up to a gauge tranformation which is fixed by choosing "London gauge") since the magnetic field is defined through
Additionally, accroding to Ampere's law
, one may derive that:
On the other hand, since
, we have
, which leads to the spatial distribution of magnetic field obeys :
with penetration depth
. In one dimesion, such
Helmholtz equation has the solution form
Inside the superconductor
, the magnetic field exponetially decay, which well explains the Meissner effect. With the magnetic field distribution, we can use Ampere's law
again to see that the supercurrent
also flows near the surface of superconductor, as expected from the requirment for intepreting
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 pefect conductor is that pefect conductor does not exhibit Meissner effect for