# Do all the electrons form Cooper pairs at absolute zero?

At $T=0K$, do all the electrons form Cooper pairs, or just the electrons near the Fermi Surface do?

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Do you mean all electrons in the conduction band? –  John Rennie Jul 16 '14 at 8:25
If you mean by $T=0\text{ K}$ absolute zero, then it's impossible to reach that temperature –  Gigi Butbaia Jul 16 '14 at 8:47
Sorry, maybe I didn't ask clearly. I am not emphasizing T=0K, but the latter part: we know for a metal, at T=0K, in the momentum space, the electrons occupy the energy states below the Fermi energy, forming a Fermi sphere. My question is in superconductor, do all the electrons form pairs (e.g. including the electrons far below the Fermi sphere) or only the electrons near the Fermi sphere surface form pairs? –  user43713 Jul 16 '14 at 19:55
Yes, I mean all the electrons in the conduction band. –  user43713 Jul 16 '14 at 20:32
Short answer: no ! But all conduction electrons are in principle all forming Cooper pairs at zero-temperature. Of course this is valid only in the thermodynamics limit. If you have only few conduction electrons, and an odd number of them, it is clear that non all of them will form Cooper pairs, since one will always remain at the end. –  FraSchelle Jul 18 '14 at 14:53

# Introduction (BCS Theory):

Cooper pairs in BCS theory are explained like this: the energy that can break cooper pairs in material is for example $10^{-4}\text{ }eV$ and thermal energy of that material is $E=kT$ where $k$ is Boltzman's Constant, so if thermal energy is lower than the energy that can break cooper pairs (in this example $kT<10^{-4}\text{ }eV$) then electrons form pairs called cooper pairs and can flow freely (forever) without any resistance, resistance decreases as temperature goes down until it reaches critical point ($T_c$) where resistance is exactly zero (this phenomenon is called Superconductivity), here's graph of that:

so if you decreased kinetic energy of molecules (temperature) one by one below critical temperature $$T_c=\frac{E_{min}}{k}$$ then all electrons that are flowing come in pairs (cooper pairs)

# Conclusion:

So answer is that all electrons that are flowing form cooper pairs (and you don't have to cool it down to $T= 0 \text{ Kelvin}$ (absolute zero which is impossible) to get cooper pairs, decreasing all molecules kinetic energy (decresing temperature) below critical temperature ($\frac{mv^2}{2} = kT\Rightarrow v = \sqrt{\frac{2kT}{m}}$ where $v$ is molecule's velocity) is enough to form cooper pairs), heres how it looks like:

Flow of electrons is called current but in superconductor current can flow freely without any resistance and current that flows is called supercurrent, In practice you can measure current of material (which is supercooled below it's critical temperature $T_c$) and calculate number of electrons from this equation: $$I = \frac{\partial q}{\partial t} = \dot q$$ where $q=ne$, (where $n$ is number of particles and $e$ is elementary charge) $$ne=\int I\text{ }dt\Rightarrow n=\frac{1}{e}\int I\text{ }dt$$

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Are you sure if $kT<10eV$, the electrons form pairs? because $10eV$ corresponds to a very high energy scale (keep in mind the room temperature T=300K is only 0.026eV. –  user43713 Jul 16 '14 at 20:01
thanks for mentioning that @user43713 , it actually should be $\approx 10^{-4}\text{ eV}$ –  Gigi Butbaia Jul 16 '14 at 20:08