In Physical, the theory of Ginzburg-Pram is a phenomenologic Théorie Supraconducteur S, proposed in 1950 by the Soviet physicists V.L. Ginzburg and L.D. Landau.

It bases on older work of L.D. Landau (1938) on the transitions from phase of the second order. This theory uses a Paramètre of order $\backslash \; psi$ called “function of wave of the electrons condensed” by Landau and Ginzburg. This parameter of order measures the Brisure of symmetry U (1) in the superconductive state.

Free energy

With this parameter of order, Pram and Ginzburg built a free energy variational having the symmetry of the phase of high temperature. This variational energy is written:

$F\_\; \{VAr\}\; =\; \backslash \; int\; D\; \{\backslash \; mathbf\; \{R\}\}\; \backslash \; left\; \backslash \; mid\; (\backslash \; frac\; \backslash \; hbar\; I\; \backslash \; nabla\; -\; e^*\; \backslash \; mathbf\; \{has\})\; \backslash \; psi\; \backslash \; mid^2\; +\; \backslash \; frac\; \{has\; (T-T\_c)\}\{2\}\; \backslash \; mid\; \backslash \; psi\; \backslash \; mid^2\; +\; \backslash \; frac\; B\; 4\; \backslash \; mid\; \backslash \; psi\; \backslash \; mid^4\; +\; \backslash \; frac\; \{(\backslash \; nabla\; \backslash \; times\; \backslash \; mathbf\; \{has\})\; ^2\}\; \{2\; \backslash \; driven\}\; \backslash \; right$

According to the principles of the theory of Pram of the transitions from phase of second order, this variational energy must be minimized compared to the variational parameters $\backslash \; psi$ and $\backslash \; mathbf\; \{has\}$.

The first equation of minimization makes it possible to obtain the superfluid density:

$\backslash \; rho\_s\; =\; \backslash \; mid\; \backslash \; psi\; \backslash \; mid^2$.

The second equation gives again the equation of Maxwell-Amp with the following definition for the current:

$\backslash \; mathbf\; \{J\}\; =\; \backslash \; frac\; \{e^*\; \backslash \; hbar\}\; \{2\; I\; m^*\}\; (\backslash \; psi^*\; \backslash \; nabla\; \backslash \; psi\; -\; \backslash \; psi\; \backslash \; nabla\; \backslash \; psi^*)\; -\; \backslash \; frac\; \{\backslash \; rho\_s\; (e^*)\; ^2\}\; \{m^*\}\; \backslash \; mathbf\; \{has\}$

It is seen that for a uniform parameter of order, one finds the equation of London and thus the Effet Meissner (expulsion of the magnetic field by the superconductor).

Length of coherence

In the absence of Magnetic field, the first equation of minimization is written a non-uniform parameter of order in the case of:

$-\; \backslash \; frac\; \{\backslash \; hbar^2\}\; \{2m^*\}\; \backslash \; nabla^2\; \backslash \; psi\; +\; has\; (T-T\_c)\; \backslash \; psi\; +\; B\; |\backslash \; psi|^2\; \backslash psi\; =0$

For a uniform state $\backslash \; nabla\; \backslash \; psi=0$, one finds:

$|\backslash \; psi|\_\; \backslash \; infty=\; \backslash \; sqrt\; \{\backslash \; frac\; \{has\; (T\_c\; -\; T)\}\{B\}\}$

One can seek a non-uniform solution of the equation of Ginzburg Pram, depending only on one coordinate $x$ and such as $\backslash \; psi\; (0)\; =0$ and $\backslash \; lim\_\; \{X\; \backslash \; to\; \backslash \; infty\}\; \backslash \; psi\; (X)\; =|\backslash \; psi|\_\; \backslash \; infty$. This solution is written in the form:

$\backslash \; psi\; (X)\; =|\backslash \; psi|\_\; \backslash \; infty\; F\; (X\; \backslash \; xi)$, where $\backslash \; xi$

is the length of coherence, and:

$F$ - F +f^3=0

It is found that the length of coherence varies as

$\backslash \; xi=\; \backslash \; sqrt\; \{\backslash \; frac\; \{\backslash \; hbar^2\}\; \{2m^*\; has\; (T-T\_c)\}\}$

Length of penetration of the magnetic field

The equation of London is written:

$\backslash \; nabla^2\; \backslash \; mathbf\; \{has\}\; =\; \backslash \; mu\_0\; \backslash \; frac\; \{(e^*)\; ^2\; \backslash \; rho\_s\}\; \{m^*\}\; \backslash \; mathbf\; \{has\}$

One introduces the length of penetration $\backslash \; lambda$ by:

$\backslash \; lambda=\; \backslash \; sqrt\; \{\backslash \; frac\; \{m^*\}\; \{\backslash \; mu\_0\; (e^*)\; ^2\; \backslash \; rho\_s\}\}$

And it is seen that the solutions of the equation of London are form $\backslash \; mathbf\; \{has\}\; (X)\; =\; \backslash \; mathbf\; \{has\}\; (0)\; e^\; \{-\; X\; \backslash \; lambda\}$, which involves that the magnetic field is different from zero only in one layer thickness $\backslash \; lambda$ close to surface to the superconductor.

By using the expression of $\backslash \; rho\_s$, one shows that $\backslash \; lambda$ varies with the Température like:

$(T-T\_c)\; ^\; \{-\; 1/2\}$

and thus that the report/ratio $\backslash \; kappa=\; \backslash \; lambda/\backslash \; xi$ is independent of the temperature.

Superconductors of the type I and II

The equation of Pram-Ginzburg as makes it possible to predict as there exist two types of Supraconducteur S, the superconductors of the type I in which the length of cicatrization of the parameter of order is larger than the length of penetration of the magnetic field (), and who return in a normal state beyond a field criticizes $H\_c$, and superconductors of the type II where the length of cicatrization is low in front of the length of penetration of the magnetic field ($\backslash \; kappa>1/\backslash \; sqrt\; \{2\}$).

Superconductors of the type I

In the superconductors of the type I, beyond a critical Magnetic field $H\_c$, supraconductivity is destroyed in all the sample which returns in a normal state.

In a superconductor of the type I, the critical field is of a few hundred Gauss what prohibits any electrotechnical application.

Superconductors of the type II

In a superconductor of the type II, when the magnetic field exceeds a value $H\_\; \{c1\}$, it is formed Vortex (linear defects the length whose the parameter of order $\backslash \; psi$ is cancelled) where a normal metal heart lets pass the magnetic field, while around of this field there exists a swirl of superconductive current which prevents the penetration of the magnetic flux in the remainder of material.

A remarkable characteristic of the vortices is that they carry a quantum of flow $\backslash \; frac\; \{H\}\; \{e^*\}$ because of the univocal character of the phase of the parameter of order $\backslash \; psi$.

The solution of the equation of Ginzburg-Pram describing the vortex also appears in the context of the theory of the fields under the name of “Nielsen-Olsen string”.

Network of vortex in the superconductors of the type II

By using the equation of Ginzburg Pram, A.A. Abrikosov established that the vortices would form a hexagonal Réseau because of the repelling powers create between vortex by the superconductive currents.

This network of vortex to the top of the field $H\_\; \{c1\}$ can be highlighted by the experiments of decoration of Bitter where magnetic particles are projected on the surface of the superconductive sample. The particles are attracted where the Magnetic field most extremely i.e. at the place is where the end of the vortices touches the surface of the sample.

Finally for a field even more extremely $H\_\; \{c2\}$, which can also be calculated by the equation of Pram-Ginzburg, supraconductivity is destroyed. This field $H\_\; \{c2\}$ can be about the Tesla with the result that the superconductors of the type II can be used in reels intended to generate intense magnetic fields.

It should be noted that for

The effect of the defects on the vortex is represented by a $T\_c$ which depends explicitly on the position. It is énergétiquement more advantageous to place the vortices where $T\_c$ is weaker, the loss of energy of condensation being less.

Relation between the theory of Pram-Ginzburg and theory BCS

L.P. Gork' OV established by methods of Fonction of Green that the theory of Ginzburg-Pram could be obtained starting from the Théorie BCS with the help of certain approximations.

The calculation of Gor' kov as makes it possible to show as $e^*=2e$, $m^*=2m$, i.e. the “condensed electrons” of Pram and Ginzburg are in fact of the even of electrons.

Pram-Ginzburg depend on time

There exist modifications of the theory of Pram and Ginzburg (Pram-Ginzburg depend on time) which make it possible to describe the dynamics of the vortices.

References

• Mr. Tinkham Introduction to Superconductivity (McGrawHill)
• A.A. Abrikosov, L.P. Gor' kov and I.E. Dzialoshinskii Methods off Quantum Field Theory in Statistical

Physics (Dover)

• P.G. Of Genoa Superconductivity off metals and alloys (Addison-Wesley)
• P.W. Anderson BASIC Concepts off Condensed Matter Physics (Addison Wesley)

• P. Mangin '' Cours of the school of the mines of Nancy ''

• V.L. Ginzburg and A.A. Abrikosov '' on the site of the Foundation Nobel ''
• E.H. Brandt '' Article of review on the superconductors of the type II ''
• '' Images of networks of vortex ''

Random links:

Equivocal of glassmaker | Large Odalengo | Protectores | Championship of Turkey of basketball | Riom-be-mountains