Just a general question, can cooper pairs be explained using the casimir effect?
No - Cooper pairs are bound by phonons.
.... in conventional superconductors.
Casimir effect is extremely weak! No way can it form such coupling strength we see not only in conventional superconductors, but also high-Tc cuprates. Besides, there's no explanation for it to suddenly kick in at Tc via a phase transition-like phenomenon.
Phase transition and sudden kick:
Well, if you say that resistivity appears as soon as 1 electron in 10.000 is unpaired or hot or out-of-crystalline-order or any other effect, then you get a very sudden kick from any kind of transition that would be smooth for each single electron, like a standard Fermi statistics.
In other words, the probability of an electron being at 5+ sigma is MUCH lower than at 5 sigma. This slope is steeper at 5 sigma than at 1 sigma.
Within such an explanation, the transition energy for a single electron (or a pair if you prefer, this is a separate question) must be several times higher than the superconductor's critical temperature.
I strongly believe this is the fundamental reason for resistivity to appear so brutally over a narrow temperature span.
True, but I don't think this has anything to do with a phase transition. What would be the order parameter?
Er.. a "phase transition" is a collective behavior, not the behavior of "one electron". This is certainly true for a superconductor. You can't have just one electron (or two, or three, etc) undergoing such a transition.
The rest of your post, I don't understand. Note that I can make the width of the transition region change by changing either the impurity of the material, the crystalline order of the material, putting magnetic atoms, etc. Not only that, the transition region here also in the magnetic susceptibility measurement, which is a more stringent test of a superconductor than simply the resistivity measurement. So it isn't just a matter of electrical transport transition.
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