dear Dolgopolov,

the presence of "spatial" ground states does not necessarily ensures the pairing of electrons.

for simplicity, consider a system with only two electrons: one on the ground state, and the other one elsewhere. a relaxation of the system by decreasing the temperature for instance does not necessarily allow the second electron to end up in the ground state. that electron must have an opposite spin with respect to the spin of the electron already in the ground state, before the pairing. otherwise, that second electron will end up on the state just above the ground's one. the whole system therefore becomes a triplet with no possibilities of pairing according the "pauli exclusion principle".

but the pairing mechanism in superconductivity is deeper than this simple fact. a coupling with phonons is necessary to keep the singlet state in the structure. this is one of the reasons for which not all materials are superconductors even at low temperature.

Dear Mohamed,

thank you for the answer.

I agree with you - the presence of spatial ground states is insufficient for the singlet pairing (sample - orthohelium, where two electrons form a triplet instead of singlet). I wonder - can spatial ground states in principle lead to singlet pairing in superconductors? In other words: Can large spatial ground states in crystals host singlets?

Bound electrons must be treated as EM flux. We all know that currents in a conductor are transported by polarization of orbits and electrons are a residual (proportional) effect only.

In a superconductor the EM flux of electrons expands its orbit form the atomic radius to the mechanical radius (as already Hirsch explained). Only this can physically explain why magnetic field lines no longer can penetrate a SC as these only can join the expanded EM flux.

The more interesting question is what kind of electron EM orbits can promote SC. Basically spin pairing of outer electrons delivers about 11eV (See 4-He) this of course is also orbit (radius) dependent. Be also aware that electron EM flux normally is 3D symmetric and after spin (EM flux) pairing one plane contains more energy what explains the Maissner (momentum) effect.

So the work function for SC promoting electrons must be < (11/2) eV. Here many Lanthanides and large Z atoms are good candidates. The counter atoms must force a regular (best cubic centered - honeycombed) crystal structure.

This as input form basic atomic physics.

Dear Jürg,

Thank you for your comprehensive explanation. However, the question is posed much more specifically: can a local eigenstate in crystal be occupied by two electrons, which form, thus, a singlet pair? If YES - is it applicable for superconductors?

Stanislav Dolgopolov : can a local eigenstate in crystal be occupied by two electrons, which form, thus, a singlet pair?

I can only say what others recently did discuss. 4-He (6-Li S electrons) is the best example of a spin paired electron, but others claim that you can also model it as two different states with a small difference in radius.

These discussions are all based on ancient physics models. Charge based QM is only a first order approximation with a huge error margin for states n=0,1,2 as here (n=1,2) you have EM wave like binding of electrons mixed with coulomb like binding. So the QM charge like approach is only sound for n>2. As a spin bond delivers 11eV you easily can get to this level for certain elements with mixed orbits.

So the answer to your question basically is yes, but you must find a material that allows for it. But I know of no QM solution that uses the proper weak force constant to model higher n states spin paired orbits. Here you are doing new physics.

(Once more: Spin bonds are magnetic/EM flux bonds on toroidal orbits.)

dear Wyttenbach,

it is a bit crude to consider electrons only as a residual proportional effect.

for sure a high level of accuracy is achieved by treating electrons from the electromagnetism formalism, leading to "em" fields.

but like in any field theory, well identified particles can emerge as particular oscillations of the field at hand.

although the setup of a polarization for these oscillations allows the consideration of fluxes, the existence of the corresponding particles as discret entities is not discarded.

as a matter of fact, light is just a particular oscillation of an "em" field as a solution of the Maxwell's equations free of charge and current, but single photons can be produced one by one for entanglement experiments for instance.

regards,

Dear Jürg, dear Mohamed,

maybe I should explain what I mean as a local eigenstate in crystals. It is not an atom orbital of higher n. It is an orbital, which may contain many atoms of crystal. Examples:

1. every conduction electron is a wave packet localized in the whole crystal, covering a huge number of atoms;

2. an impurity can create local states covering many neighboring lattice sites;

3. in solid hydrogen every electron state contains two H atoms;

4. standing states on Brillouin zone edge are also limited in space due to constructive multiple electron reflection on many atoms;

5. weak and strong Anderson's localisation due to lattice disorder.

All these local states can host electron singlets. Can it be considered as the pairing mechanism ?

Stanislav Dolgopolov : All these local states can host electron singlets. Can it be considered as the pairing mechanism ?

SC is/must be a global effect, at least if you look for commercial use. Key is to promote global EM flux orbits = closed magnetic field lines. Impurities could prevent such orbits but also promote spin pairs forming/adding flux to such orbits. This then looks like a roulette game with 10'000 tries an one hit!

dear Dolgopolov,

theoretically, the absolute zero temperature is not affordable. meaning that all structures vibrate.

no matter the number of local states hosting the singlet pairs that have been formed from whatever reasons, they are most likely subject to a destruction due to the vibration of the structure.

for superconductivity, the current valid theory, namely "bcs" suggests that it is the vibration of the structure that must create local states for hosting these singlet pairs.

once it is done, the system transit from a system of electrons to a system of "cooper pairs".

the new quantum particles are no longer fermions, but bosons. meaning that they a synergetic effect since they are allowed to somehow "merge".

it is like a state of resonance with a wave function so powerful that almost nothing can influence the collective behavior of the corresponding particles.

but as the temperature grows, the vibration becomes more important leading to the breaking of the pairs: the system comes back to a system of fermions again. this is the main reason for which high temperature superconductivity is still incomprehensible.

but for low temperature superconductivity, the "bcs" theory works quite well. it is all about finding the right structures of the material.

wish you can have further clarity now.

regards,

Dear Mohamed,

you're right - sufficiently strong thermal fluctuations can lead to the destruction of local states, that is the electrons leave local states, the pairs break. However, if the local state is limited in space, then the delocalization temperature is not arbitrarily small, so below a certain T the singlet pair is lasting. There are many samples of lasting pairs in local states: most atoms and molecules are stable due to stability of their localized singlet pairs. Insulating materials are insulating because their valence electrons (usually singlets) cannot hop from atom to atom. So we can expect that any local states in crystal also may be stable. One fact confirms our expectation: many superconductors become insulating just above Tc. BCS cannot explain this fact (and many others).

Regards, Stanislav