Why does the model interaction for Cooper pairs make sense?

[sam_bell]

Hi All,

I am trying to understand why Cooper pairs form. The textbook proof that I am reading (Grosso & Parravicini) starts out as follows: Imagine a 'passive' electron gas filled out to some Fermi sphere kF. Add two 'extra' electrons that interact via a potential U(x-x'). Transform to k-space and take the interaction U_kk' to be attractive within a narrow window of width hω_D/2pi above the Fermi sphere. This is the part where I don't follow. Why is this reasonable? Why does this make sense?

Any insights or helpful leads would be appreciated.

Thanks,
Sam

 

[Simon Bridge]

Do you mean, "why posit any attraction at all?"

 

[sam_bell]

No, sorry, I would like to understand why the form given the attraction makes sense. It's beginning to make more sense to me. Would it be fair to say .. The interaction U_kk' for k < kFermi is not operative because of the filled Fermi sea, and not for |Ek-Ek'| >> hω_D because no phonons are available to carry the energy discrepancy. Then, for simplicity, we take U_kk' = -U0 (a constant) within the window |Ek - Ek'| < hω_D and zero elsewhere.

 

[Simon Bridge]

That's right - remember the context is of very low temperatures so all the lower energy levels are filled.

Note: the author is simplifying the situation in the hopes that this will help you imagine the situation.
If you continue to have trouble with this, you may be well advised to find another author who uses a different approach.