|
|
| Line 1: |
Line 1: |
| [[File:EfektMeisnera.svg|thumb|400 px|right|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 author is known as Irwin. Doing ceramics is what my family and I enjoy. Hiring is her working day occupation now but she's always needed her own company. His family life in South Dakota but his spouse desires them to transfer.<br><br>Feel free to surf to my webpage :: [http://www.ddaybeauty.com/node/14766 ddaybeauty.com] |
| | |
| The '''London equations''', developed by brothers [[Fritz London|Fritz]] and [[Heinz London]] in 1935,<ref>{{cite doi|10.1098/rspa.1935.0048}} | |
| </ref>
| |
| 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.<ref>
| |
| {{cite book
| |
| | author = Michael Tinkham
| |
| | title = Introduction to Superconductivity
| |
| | publisher = McGraw-Hill
| |
| | year = 1996
| |
| | isbn = 0-07-064878-6}}</ref><ref>{{cite book
| |
| | author = Neil W. Ashcroft
| |
| | coauthors = N. David Mermin
| |
| | title = Solid State Physics
| |
| | publisher = Saunders College
| |
| | year = 1976
| |
| | isbn = 0-03-083993-9
| |
| | page = 738}}</ref><ref>{{cite book
| |
| | author = Charles Kittel
| |
| | title = Introduction to Solid State Physics
| |
| | publisher =
| |
| | year = 1999
| |
| | isbn = 0-47-141526-X}}</ref>
| |
| A major triumph of the equations is their ability to explain the [[Meissner effect]],<ref>{{cite journal
| |
| |last= Meissner
| |
| |first= W.
| |
| |title=Ein neuer Effekt bei Eintritt der Supraleitfähigkeit
| |
| |coauthors= R. Ochsenfeld
| |
| |journal= Naturwissenschaften
| |
| |volume= 21
| |
| |year= 1933
| |
| |doi= 10.1007/BF01504252
| |
| |pages= 787 |bibcode = 1933NW.....21..787M
| |
| |issue= 44 }}
| |
| </ref>
| |
| 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:
| |
| :<math>\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}. </math>
| |
| Here <math>{\mathbf{j}}_s</math> is the superconducting [[current density]], '''E''' and '''B''' are respectively the electric and magnetic fields within the superconductor,
| |
| <math>e\,</math> <!-- do not delete "\'": It improved display of the formula in certain browsers. --->
| |
| is the charge of an electron & proton, | |
| <math>m\,</math> <!-- do not delete "\'": It improved display of the formula in certain browsers. --->
| |
| is electron mass, and
| |
| <math>n_s\,</math> <!-- do not delete "\'": It improved display of the formula in certain browsers. --->
| |
| is a phenomenological constant loosely associated with a number density of superconducting carriers.<ref name="James F. Annett 2004 58">{{cite book
| |
| | author = James F. Annett
| |
| | title = Superconductivity, Superfluids and Condensates
| |
| | publisher = Oxford
| |
| | year = 2004
| |
| | isbn = 0-19-850756-9
| |
| | page = 58}}</ref>
| |
| Throughout this article [[Gaussian units|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"<ref name="James F. Annett 2004 58"/><ref>{{cite book
| |
| | author = John David Jackson
| |
| | title = Classical Electrodynamics
| |
| | publisher = John Wiley & Sons
| |
| | year = 1999
| |
| | isbn = 0-19-850756-9
| |
| | page = 604}}</ref>
| |
| in terms of the [[vector potential]] '''A''':
| |
| :<math>\mathbf{j}_s =-\frac{n_se^2}{mc}\mathbf{A}. </math>
| |
| | |
| 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.<ref>
| |
| {{cite book
| |
| | author = Michael Tinkham
| |
| | title = Introduction to Superconductivity
| |
| | publisher = McGraw-Hill
| |
| | year = 1996
| |
| | isbn = 0-07-064878-6
| |
| | page = 6}}</ref>
| |
| | |
| ==London Penetration Depth==
| |
| | |
| If the second of London's equations is manipulated by applying [[Ampere's law]],<ref>(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''.)</ref>
| |
| :<math>\nabla \times \mathbf{B} = \frac{4 \pi \mathbf{j}}{c}</math>,
| |
| then the result is the differential equation
| |
| :<math>\nabla^2 \mathbf{B} = \frac{1}{\lambda^2}\mathbf{B}, \qquad \lambda \equiv \sqrt{\frac{m c^2}{4 \pi n_s e^2}}. </math>
| |
| Thus, the London equations imply a characteristic length scale, <math>\lambda</math>, 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
| |
| | |
| :<math>B_z(x) = B_0 e^{-x / \lambda}. \,</math><!--- Do not delete "\'". It serves to improve the display of this equation in certain browsers ---> | |
| | |
| 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,<ref name="Michael Tinkham 1996 5">{{cite book
| |
| | author = Michael Tinkham
| |
| | title = Introduction to Superconductivity
| |
| | publisher = McGraw-Hill
| |
| | year = 1996
| |
| | isbn = 0-07-064878-6
| |
| | page = 5}}</ref>
| |
| 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]]
| |
| :<math>\mathbf{F}=e\mathbf{E}+ \frac{e}{c} \mathbf{v} \times \mathbf{B}</math> | |
| 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]],
| |
| :<math>\nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}}{\partial t}</math>, | |
| to obtain
| |
| :<math> \frac{\partial}{\partial t}\left( \nabla \times \mathbf{j}_s + \frac{n_s e^2}{m c} \mathbf{B} \right) = 0.</math>
| |
| | |
| 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.<ref>{{cite book
| |
| | author = John David Jackson
| |
| | title = Classical Electrodynamics
| |
| | publisher = John Wiley & Sons
| |
| | year = 1999
| |
| | pages = 603–604
| |
| | isbn = 0-19-850756-9}}</ref><ref>{{cite book
| |
| | author = Michael Tinkham
| |
| | title = Introduction to Superconductivity
| |
| | publisher = McGraw-Hill
| |
| | year = 1996
| |
| | pages = 5–6
| |
| | isbn = 0-07-064878-6}}</ref>
| |
| Current density is defined according to the equation
| |
| :<math>\mathbf{j}_s = n_s e \mathbf{v}.</math>
| |
| 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
| |
| :<math>\mathbf{v} = \frac{1}{m} \left( \mathbf{p} - \frac{e}{c}\mathbf{A} \right) </math>
| |
| is defined by dividing the gauge-invariant, kinematic momentum operator by the particle mass ''m''.<ref>{{cite book
| |
| | author = L. D. Landau and E. M. Lifshitz
| |
| | title = Quantum Mechanics- Non-relativistic Theory
| |
| | publisher = Butterworth-Heinemann
| |
| | year = 1977
| |
| | pages = 455–458
| |
| | isbn = 0-7506-3539-8}}</ref>
| |
| We may then make this replacement in the equation above. However, an important assumption from the [[Bcs theory|microscopic theory of superconductivity]] is that the superconducting state of a system is the ground state, and according to a theorem of Bloch's,<ref name="Michael Tinkham 1996 5"/>
| |
| in such a state the canonical momentum '''p''' is zero. This leaves
| |
| :<math>\mathbf{j}_s =-\frac{n_se_s^2}{mc}\mathbf{A}, </math>
| |
| which is the London equation according to the second formulation above.
| |
| | |
| ==References==
| |
| <references />
| |
| | |
| [[Category:Superconductivity]]
| |
| [[Category:Equations]]
| |