BCS theory, Richardson model and Superconductivity I'm studying Richardson Model in second quantization. There are many initial points that I don't understand:


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*We supposed that an attractive force between 2 electrons exists, due to electron-phonon interaction: an electron generates a phonon that is absorbed by the other electron, and we prove that the resulting potential is attractive. But is this really stronger than the Coulomb repulsive potential? How can one estimate this?

*Why does it neglect the attractive force between electrons with opposite spin? ($\pm$) They have an opposite magnetic momentum too...

*Why do we only consider the attractions between electrons with opposite spin?

*Why do we suppose $k_f ~R ~\ll ~1$ , where $k_f$ is (Fermi momentum)/$\hbar$, and $R$ the interaction distance? 
 A: I will just answer the first part of question: is phonon attraction stronger than Coulomb?
Short answer: No.
Longer answer: Nothing (in condensed matter) is ever stronger than the Coulomb force.
Longest answer: There are two aspects to consider. First is the self-screening of the electrons, which will add a mass term to the photons, giving a Yukawa-esque potential ($e^{-mr}/r$). Second, although we always work in momentum space, it's actually unhelpfully abstract in this case. You need to imagine what the force looks like in real space and time. The Coulomb force is a 1/r^2 instaneous repulsion, but it decays very fast (instaneous) in time. The screened Yukawa/Thomas-Fermi force decays less slowly (exponentially with scale $m$) in time. The phonon attraction, however, is a retarded force, meaning it only acts some finite time after the electron has left. Thus if you imagine a plot of the potential as as function of spacetime, there is a singularity at the origin, but there is a small dip after it! In terms of dynamics this is even easier to understand: electrons cause small lattice deformations as they move around, but the lattice deformations need time to be set up and relax again, and this time-scale is larger than the time it takes an electron to get across the distortion; thus another electron sees a small, attractive hole into which it could fall.
Because the Fermi liquid is so unstable (it has too many low energy degrees of freedom) any attractive interaction in any form will push it towards a broken symmetry state, such as superconductivity (or charge density waves, etc.). 
So although the phonon attraction isn't "stronger", it can beat Coulomb repulsion. It just has to be clever about it.
