London equations

Last updated
As a material drops below its superconducting critical temperature, magnetic fields within the material are expelled via the Meissner effect. The London equations give a quantitative explanation of this effect. EfektMeisnera.svg
As a material drops below its superconducting critical temperature, magnetic fields within the material are expelled via the Meissner effect. The London equations give a quantitative explanation of this effect.

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 London American physicist

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.

Meissner effect Expulsion of a magnetic field from a superconductor during its transition to the superconducting state

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.

Contents

Formulations

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 & proton, is electron mass, and 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.

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]

,

then the result is the differential equation

Thus, the London equations imply a characteristic length scale, , 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

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

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,

,

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.

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

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 defined by dividing the gauge-invariant, kinematic momentum operator by the particle mass m. [13] Note we are using 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

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

Related Research Articles

Lorentz force mutual force exerted by two punctual charges in relative motion

In physics the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force of

Superconductivity physical phenomenon

Superconductivity is the set of physical properties observed in certain materials, wherein electrical resistance vanishes and from which magnetic flux fields are expelled. Any material exhibiting these properties is a superconductor. Unlike an ordinary metallic conductor, whose resistance decreases gradually as its temperature is lowered even down to near absolute zero, a superconductor has a characteristic critical temperature below which the resistance drops abruptly to zero. An electric current through a loop of superconducting wire can persist indefinitely with no power source.

In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Lazarevich Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. Later, a version of Ginzburg–Landau theory was derived from the Bardeen–Cooper–Schrieffer microscopic theory by Lev Gor'kov, thus showing that it also appears in some limit of microscopic theory and giving microscopic interpretation of all its parameters. The theory can also be given a general geometric setting, placing it in the context of Riemannian geometry, where in many cases exact solutions can be given. This general setting then extends to quantum field theory and string theory, again owing to its solvability, and its close relation to other, similar systems.

Josephson effect quantum physical phenomenon

The Josephson effect is the phenomenon of supercurrent, a current that flows indefinitely long without any voltage applied, across a device known as a Josephson junction (JJ), which consists of two or more superconductors coupled by a weak link. The weak link can consist of a thin insulating barrier, a short section of non-superconducting metal (S-N-S), or a physical constriction that weakens the superconductivity at the point of contact (S-s-S).

Magnetic moment extensive physical property

The magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include: loops of electric current, permanent magnets, elementary particles, various molecules, and many astronomical objects.

Magnetic potential Integral of the magnetic field

The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, or simply vector potential, A; and the magnetic scalar potentialψ. Both quantities can be used in certain circumstances to calculate the magnetic field B.

In electromagnetism, the Lorenz gauge condition or Lorenz gauge is a partial gauge fixing of the electromagnetic vector potential. The condition is that This does not completely determine the gauge: one can still make a gauge transformation where is a harmonic scalar function.

The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction, e.g. Coulomb. The equation was first suggested for description of plasma by Anatoly Vlasov in 1938 and later discussed by him in detail in a monograph.

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:

Maxwell stress tensor

The Maxwell stress tensor is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impossibly difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand.

Micromagnetics is a field of physics dealing with the prediction of magnetic behaviors at sub-micrometer length scales. The length scales considered are large enough for the atomic structure of the material to be ignored, yet small enough to resolve magnetic structures such as domain walls or vortices.

In superconductivity, the superconducting coherence length, usually denoted as , is the characteristic exponent of the variations of the density of superconducting component.

In plasma physics, the Hasegawa–Mima equation, named after Akira Hasegawa and Kunioki Mima, is an equation that describes a certain regime of plasma, where the time scales are very fast, and the distance scale in the direction of the magnetic field is long. In particular the equation is useful for describing turbulence in some tokamaks. The equation was introduced in Hasegawa and Mima's paper submitted in 1977 to Physics of Fluids, where they compared it to the results of the ATC tokamak.

Mathematical descriptions of the electromagnetic field Formulations of electromagnetism

There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

The Chandrasekhar number is a dimensionless quantity used in magnetic convection to represent ratio of the Lorentz force to the viscosity. It is named after the Indian astrophysicist Subrahmanyan Chandrasekhar.

Type-1.5 superconductors are multicomponent superconductors characterized by two or more coherence lengths, at least one of which is shorter than the magnetic field penetration length , and at least one of which is longer. This is in contrast to single-component superconductors, where there is only one coherence length and the superconductor is necessarily either type 1 or type 2. When placed in magnetic field, type-1.5 superconductors should form quantum vortices: magnetic-flux-carrying excitations. They allow magnetic field to pass through superconductors due to a vortex-like circulation of superconducting particles. In type-1.5 superconductors these vortices have long-range attractive, short-range repulsive interaction. As a consequence a type-1.5 superconductor in a magnetic field can form a phase separation into domains with expelled magnetic field and clusters of quantum vortices which are bound together by attractive intervortex forces. The domains of the Meissner state retain the two-component superconductivity, while in the vortex clusters one of the superconducting components is suppressed. Thus such materials should allow coexistence of various properties of type-I and type-II superconductors.

Flux pumping is a method for magnetising superconductors to fields in excess of 15 teslas. The method can be applied to any type II superconductor and exploits a fundamental property of superconductors. That is their ability to support and maintain currents on the length scale of the superconductor. Conventional magnetic materials are magnetised on a molecular scale which means that superconductors can maintain a flux density orders of magnitude bigger than conventional materials. Flux pumping is especially significant when one bears in mind that all other methods of magnetising superconductors require application of a magnetic flux density at least as high as the final required field. This is not true of flux pumping.

Chandrasekhar-Kendall functions are the axisymmetric eigenfunctions of the curl operator, derived by Subrahmanyan Chandrasekhar and P.C. Kendall in 1957, in attempting to solve the force-free magnetic fields. The results were independently derived by both, but were agreed to publish the paper together.

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.
  3. Neil Ashcroft; David Mermin (1976). Solid State Physics . Saunders College. p. 738. ISBN   0-03-083993-9.
  4. 1 2 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. 1 2 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. 1 2 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.