London equations

The London equations (named after the brothers Fritz and Heinz London) are based on a postulate and replace the Ohm's law in a superconductor. You describe so well how the magnetic field behaves in such material. One result is approximately that the magnetic field despite contrary predictions slightly penetrates into the superconductor ( penetration depth? L ).

Experimental motivation

Due to the Meissner effect, a superconductor is an ideal diamagnetic ( magnetic susceptibility χ = -1), and its interior should be a magnetic field. This prediction can be but not experimentally confirmed. It is observed that the magnetic field of thin superconducting films is not completely pushed out, so the interior is not quite field-free. It also receives the continuity condition for the magnetic field at the edge of the superconductor.

Formulation

To explain this, one replaces the classical Ohm's law for the electric current density j and the electric field E

By the London equation:

With

• The phase of the macroscopic wave function,
• The vector potential of the magnetic field and
• N is the particle number density of charge carriers.

( Derivation of the equation see separate section. )

There are two useful transformations of this equation, which will be sometimes referred to as the 1st and 2nd London equations:

The phase S is no contribution to these two equations - to the first equation is not, because the phase and thus is location-dependent constant in time, and the second equation is not as applicable.

Warning: Although the phase fraction does not contribute to the last two formulas, it may nevertheless not be neglected! Would the phase fraction not enter, so that would mean that the current density should be no magnetic field is zero. In reality, however, the phase gradient can contribute to the current density even, then this might not be forced to zero, that is, the current density is not zero, even though no magnetic field is present. The approach of a macroscopic wave function is also made for superfluids. In this case, it is actually the phase S, which performs approximately the fountain effect quantized or vortices.

Theoretical explanation of the Meissner effect

With the help of Maxwell's equation, we can rewrite the second London equation:

The solution of this equation describes an exponential decay of the magnetic field inside the superconductor, as observed in the experiment (see Meissner effect). A homogeneous magnetic field of strength B 0 in the z direction, which bears against the surface of the superconductor ( perpendicular to the x -axis), the solution is

Thus, the magnetic field decays exponentially from the superconductor, with the penetration depth? L, in which:

Here, the mass of the electron, q is the charge, n is the number density of the superconducting charge carriers and the magnetic field constant. This gives the Abschirmstromdichte:

In a thin outer layer of the superconductor thus shielding current flows perpendicular to the magnetic field.

Derivation of the London equations on the macroscopic wave function

Approach: the superconducting state is a quantum mechanical state, which extends over macroscopic length scales. It can therefore be described by the macroscopic wavefunction:

In this case, it is assumed that a constant, real (!) Has amplitude and phase, only the S is a function of location. this corresponds to the number density of the Cooper pairs. So a constant amplitude implies a constant particle number density. This assumption is reasonable, since the Cooper pairs in the superconductor are all negatively charged and repel each other. An imbalance of the particle number density would mean an electric field, which would be immediately compensated.

The canonical momentum operator in the presence of a magnetic field is:

Applied to the wave function gives:

So:

Directly with the following:

This is the above mentioned London equation.