It is not imaginary construct. It is governed collectively by both Landau-Fermi liquid theory and BCS theory as explained below.

According to the landau-fermi liquid model In the normal phase of metal the particle-hole pair state having energy < the fermi energy is possible only if there exist however small attractive potential between the particle-particle pair( called cooper pair) but because of the small magnitude of this pairing energy as campared to high thermal energy the cooper pairs breaks in the normal metal phase.But in the superconducting phase of the metal at low temperature pairing energy domminates the thermal energy and there exist a stable state for cooper pair below fermi energy in the superconducting phase now these cooper pairs have electron of half integer spin forming the triplet state (state with integer spin) the cooper pairs act as the boson particles.Now according to the BCS theory the pairing energy leads to the gap in the continous energy spectrum of metals so there exist a low energy state to which all the cooper pair try to condenstate which in turns at low temperature leads to the Bose-Einstein condensation of these cooper pairs and almost zero resistance for the cuurent. (like a superfluid state having practicaly very small viscosity). The reason which I have found in the literature that the coloumb repulsions between the electrons is screened by this attractive potential between electrons in the metals at very low temprature which forbid the e-e sccattring (responsible for resistance) leads to the ball room dance of cooper pairs.

Cooper pairs do not have triplet state. They rather have singlet state and that is the reason that superconductors exhibit dia-magnetism. As such the remains unanswered.

Yatendra, in 'more or less normal' superconductors we have the facts that

(i) the spatial extent of a single Cooper pair is quite considerable,

(ii) especially compared to the density of Cooper pairs (-> "avaerage distance between pairs if thought to be point-like. The latter is pointless, of course :-)

This bears the result that Cooper pairs massively overlap each other in space.

We usually like to say that the Cooper pairs are individually protected against scattering by the Cooper pair binding energy but the 'real' superconducting state is a "many Cooper pair state" and its stability also deives from this fact. So, the many overlapping Cooper pairs actually form one many-body quantum state. The movement(s) of all electrons / Cooper pairs is organized in this single wave function and this object as a whole is forming the supercurrent.

The ballroom dance image is maybe of help in imagining what it it takes to have this strange, stable many particle state.

On the pages you linked to, there is also an interesting historical account of how Bardeen and Colleagues assembled insight after insight - over more than 25 years of time. Very interesting and enlightening. Thanks for the link.

Thanks Kai. Your observations are useful. It becomes clear that ball room dance is certainly an imaginary construct that has been used to explain scattering free motions of Cooper pairs. You are right that Bardeen and colleagues added many insight for better understanding of their theory. However, the model still has several points which can not be clearly understood in terms of widely reported arguments. For example why two electrons located at distances significantly larger than the distance between two nearest neighbour electrons make Copper (not the nearest neighbour electrons).

In explaining superfluidity of liquid Helium-4 Prof Leggett has some where quoted the example of crowd ( people having random position and random motions with no correlation of their steps as a representative of normal fluid) and army platoon (ordered positions in normal space, no relative motions between two soldiers with a definite correlation between their steps as a representative of p=0 condensate or superfluid state) to distinguish between He-I (normal helium-4 liquid) and He-II (superfluid Helium-4). Can we not use the same analogy to distinguish between normal and superconducting states of conduction electrons ? I would appreciate your comments.

Yatendra, can you provide a source for that saying of Leggett? Instantanously I don't feel like having to say anything in this respect. I don't think I have an appropriate pictorial representation of the superfluid phase.

Concerning the Cooper pairs above (I'd think that "Copper" is a typo there): I'd say you need to start from Bloch states. Distance is not something you could define (I'd think) for Bloch states, since any two Bloch states are constructed to be everywhere in space. But electrons in crystals - particularly in metals and near the Fermi level should be thought of Bloch-state-like electrons to first approximation. So, electron phonon interaction as in BCS theory is not "a localized event" in space, but a process (ideally) involving elementary objects (electronic quasiparticles and vibronic quasiparticles) that are defined on the entire lattice. This is also the starting point of Bardeens thinking (as I understand it from that historical account on the pages you linked to). In this case "nearest neighbour electron" would not be an appropriate concept.

Hello Prof Kai,

Following is the reference where Prof. Leggett has made the analogy and it was also pictured on the Science magazine cover.

A Leggett, International. J. Mod. Phys. B, 15 (2001),

1312-1313. Which can be downloaded from the link.

http://icmt.illinois.edu/publications/leggett/2001-AL-JOP-1312-ResponseToFeenbergMedalPresentation.pdf

A picture depicted on the cover of Science magazine of De-

cember 22, 1995, declaring BEC in TDG as the molecule of

the year.

You are right. In fact I do not think that I differ with you what you have said here. In fact when we have a many body system where particles interact basically through pair interaction and have effective attraction (say), we really find that as a result of this interaction each particle interact with N-1 other particles in the system and this reality is independent of which two particles we pick up or which individual particle we are talking about; this is seen in reality in solids and liquids. If they form a real bound pair, then such particles are found be nearest to each other not far from each other (leaving many other particles in between).

So if there is some interaction between two electrons, it is expected to occur in every two particles whether far or near. In view of this I have a conceptual difficulty, if we say that two electrons making Cooper pairs are far of from each other,. And it is for this reason that the perception of large size of a Cooper pair becomes difficult. Electrons in Cooper pairs can be visualized to be far from each other if electron near the Fermi surface are believed to form such pairs ? ?.

Similarly, I have difficulty in accepting that an electron passing through a lattice attract +ve ions and these ions get displaced towards moving electron and in this way generate/absorb a phonon. Because +ve ions are much heavier and they are strongly fixed at their position, it becomes very difficult that they get displaced from their positions towards the moving electron through a channel in the lattice; simple physical argument concludes that it is the lighter particle, electron, which can be expected to deviate from the direction of its velocity.

Bounding electrons in a superconductor can be visualised in the context of the pondero-motive force, similar to the pattern forming sand on a vibrating plate.  Being scattered by the vibrating lattice, the electrons seek a low energy pattern, where electron pairing is the most important constituent.  In short ball dance is a convenient description but may not be the complete picture.

@ Xueheng Zheng.  Thank you very much for your answer which certainly helps me understand how exactly BCS model is perceived by researchers in the field.  It appears that even the followers of BCS model differ in their perception on Cooper pairs.  While some people believe that all electrons form Cooper pairs, others believe that only those electrons form Cooper pairs which fall close to Fermi level.  In any case ever since the discovery of high temperature superconductivity, the validity of BCS theory has been seriously questioned.  In this context Hirsch has reported his detailed comments and some of your recent papers not only recognise his comments but also suggests how BCS model could be improved.  However,  I find different reasons for the failure of BCS theory.    The basic foundations of this theory have serious errors  (pl. see the attached paper).  They do not agree with certain physical realities of the state of electrons at low temperatures.    For the reasons best known to the researchers, these errors are being ignored today but I am sure that one day this fact would be recognised by all researchers in the field.