Cooper pair

As Cooper pairs of electrons in mergers special materials are referred to in the superconducting state pairs. Cooper pairs occur in metallic and ceramic materials. The phenomenon of Cooper pair formation is named after the first description in 1956 by Leon Neil Cooper and receives in the corresponding BCS theory its significance.

The same phenomenon can also occur between other particles and in other contexts, such as between two adjacent atomic nuclei in the superfluid state of 3He below a temperature of 2.6 mK .. On the other hand there are in the superfluid 4He no Cooper pairs, since the atoms bosons.

Explanation

In metals, the conduction electrons can move almost freely between the atoms. This " electron gas " consisting of fermions, and therefore subject the Fermi distribution that predicts a certain velocity distribution from zero to very high values ​​( the characteristic temperature is Kelvin). The movement of the nuclei, however, plays a relatively minor role ( the characteristic temperature is here the so-called Debye temperature of about 500 K). But only at even lower temperatures, there is a not negligible pairwise attraction of the electrons by the atomic motion. The strength ( Schwäche! ) this interaction corresponds to temperatures of only about 10 K (~ - 260 ° C); corresponding energies and lifetimes of the size. Here is the Planck constant divided by the result for corresponding typical phonon frequencies, but what still nothing proves experiments that show that it is the participating particles actually to phonons ( quantized atomic vibrations ) is, and not to other types of excited states of the system, but rather are based on the so-called isotope effect, what Leon Neil Cooper brought to the ideas presented below.

The motion of the nuclei takes place as a wave phenomenon through the whole medium to give ( after quantization ) the phonons. It takes considerable time delay, resulting in a weak polarization of the lattice results, which more than offset the Coulomb repulsion due to their higher mass. A second electron can now track this polarization lower its energy, ie it is weakly bound. The result, mediated by the lattice motion, a Cooper pair. The formation of Cooper pairs is thus based - like all polarization effects - on a weak indirect interaction: the electrons attract each other because the system is polarized by the interaction.

Can be compared under the gravity action of a first particle of this effect with the formation of a dent in a tympanic: this buckling, a second particle which is also moved to the tympanic membrane, attracted by the first, so that both are bonded together.

Since moving the two electrons involved in the opposite direction, is the total momentum of the Cooper pair ia very small or zero.

The " footprint " of each electron in a Cooper pair is described by its wave packet. If they move in the opposite direction, decay Cooper pairs, because the wave packets barely overlap, others are reformed.

Estimating the uncertainty principle the expansion of the wave packets from, one comes to values ​​of up to 10-6 m. A comparison with the average distances of the electrons in the crystal lattice results in the surprising result that the radius of the Cooper pair can be of the order given, so that at least 1010 can be other electrons between electrons of a Cooper pair. Of these, about a million other electrons so similar and overlapping wave packets that form Cooper pairs. The Cooper pairs are therefore almost as numerous as the electron itself

The main mechanism for explaining superconductivity (see below) is that, in contrast to the electrons, which, because of the Fermi statistics somehow " go out of the way " each other, condense into a coherent state, as generally for the superconductivity and is characteristic of so-called superfluids. Although the commutation of two Cooper pairs do not correspond exactly to those of Bose particles, they are in these but similar.

Importance in superconductors

Electrons belong to the group of particles of fermions and have spin 1/2 ( cf. spin- statistics theorem ). The Fermi -Dirac statistics shows that in a two- electron system without spin -orbit coupling, therefore, with symmetrical spatial wave function, the spin function must be antisymmetric, ie about This clearly means that the spin of an electron to the "top " shows (ie it is 1 / 2, units of the reduced Planck's constant), while the other spin "downward " has ( i.e., it is 1/2, in the same units ). Thus, anti- parallel alignment of the spins! The total spin of the Cooper pair is in this case 0, which corresponds to the so-called singlet states. Another, albeit less common case is the parallel alignment of the individual spins of the Cooper -pair electrons, and, therefore, adds to the total spin 1. Here, one speaks of the triplet state. Experimentally it can be shown by such experiments tunnel ( seltener! ) state, as this Cooper pairs can tunnel through larger ferromagnetic barrier ..

For both cases, the Cooper pairs are so because they have integer spin, as long as they can be considered as a single "compound particles ", not fermions, but bosons. But does not apply the Fermi -Dirac statistics, but the Bose -Einstein statistics for bosons. This states - speaking clearly - that the Cooper pairs follow a " herd mentality ", so that the above-mentioned coherent state can result in: All pairs move with the same speed in the same direction and are strictly coupled to each other

The latter auxiliary means, among other things, that the situation really should not be compared to a Bose -Einstein condensate, since the Cooper pairs can not be considered as independent particles of a Bose - gas. Nevertheless This explains the properties of metallic superconductors, since the Cooper pairs may occupy all one-and- the same quantum mechanical state as an effective Bose particles ( anti -Pauli- principle).

So you it definitely has to do with a macroscopic, collective quantum phenomenon.

Since the expansion of the wave packets of each Cooper pair is almost macroscopic size, they can by thin insulating layers tunnel ( Josephson effect). It has been experimentally demonstrated that always two electrons tunnel through the barrier.

Energy gap

Mathematically expressed, the tendency to the formation of Cooper pairs by the fact that the Hamiltonian of the system in addition to the usual bilinear terms ( with the electron creation operators and the corresponding annihilation operators ) and quadratic terms of the unusual shape occur:

 

Here k is the wave number of the electron, εk their energy in the normal state and Δ an assumed to be real pairing parameters.

Ground state and excited states of the system are changed not only quantitatively but also qualitatively by the interaction. The ground state energy E is only slightly elevated, but - what is essential - it now forms an energy gap the size of the excited states of. This has, inter alia, means that the electric resistance with a correspondingly low temperatures is everywhere zero.

References and footnotes

  • Quantum physics
  • Superconductivity
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