# Cooper pair density

Is it possible to calculate the average number of Cooper-pairs, existing at any given moment, per cubic centimeter of niobium (or YBCO), at their critical temperatures? I realize that Cooper pairs form and dissolve continuously, and constantly change partners, so perhaps its not easy to calculate - unless there's some equilibrium value for a given temperature and material.

## Answers and Replies

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DrDu
Yes, at the critical temperature, it is 0.

ZapperZ
Staff Emeritus
Is it possible to calculate the average number of Cooper-pairs, existing at any given moment, per cubic centimeter of niobium (or YBCO), at their critical temperatures? I realize that Cooper pairs form and dissolve continuously, and constantly change partners, so perhaps its not easy to calculate - unless there's some equilibrium value for a given temperature and material.
I think you are asking for the superfluid density. If this is true, then you may want to look up the Uemura relation that connects the superfluid density to temperature. The superfluid density, in turn, is typically defined as the being proportional to ns/m*, where ns is the density of superconducting electrons and m* is their effective mass.

Zz.

DrDu, ZapperZ, thanks for the replies. I gather that if I look up the Uemura relation, there might be tables, someplace on the internet, that shows the actual superfluid density, for a given material at a given temperature.

Is the effective mass "m" of the Cooper pairs simply double the electron mass?

ZapperZ
Staff Emeritus
Is the effective mass "m" of the Cooper pairs simply double the electron mass?
Er.. no. You need to know quite a bit of solid state physics here. The effective mass is the mass of the single-particle electron. It depends on the band structure.

Zz.

I believe I have found the answer to the question I posed above, from this 2003 post: http://www.phys-l.org/archives/2003/02_2003/msg00393.html A typical superconductor, ( at low temperature, when all the mobile electrons are in the superconducting state) , has a current carrying electron density of 10^22 per cubic centimeter (though it's not quite clear from the text whether this is the number of Cooper pairs or individual electrons comprising those Cooper pairs).

DrDu
That is the total valence electron density of copper.

DrDu, thank you for that clarification. Presumably at a low enough temperature, they all condense into Cooper-pairs.

I remember reading someplace, that the superconducting wire that they use in MRI machines, and other applications, consists of many individual niobium-alloy, (or whatever material they use), strands imbedded in a copper matrix. The reason stated is that the supercurrent flows on a wire's surface, (below the London penetration depth), so that it would be wasteful to have one large superconductor wire, as most of the wire's cross-sectional area, (and presumably most of the valence electrons), would not be carrying current.

I never bothered looking at the piece of niobium-tin superconducting wire I bought under a magnifying lens to see if it was constructed that way. I used to have an old microscope. If I can find it I'll take a look at the ends of my NbTi wire.

ZapperZ
Staff Emeritus
DrDu, thank you for that clarification. Presumably at a low enough temperature, they all condense into Cooper-pairs.
Actually, no they don't.

This is painfully obvious in high-Tc superconductors, where the presence of the pseudogap in the normal state has been debated as either being the precursor of superconductivity or competing with superconductivity. It has also been measured that maybe no more than 20% of the free charge carrier actually condenses into the superfluid.

So no, your estimation using that value will be way off! It isn't as easy as you think.

Zz.

ZapperZ,

I noticed that in the paper you linked to that for pure, superconducting metals it seems to be saying that all conduction electrons participate in the supercurrent, in the very first sentence. What confuses me is that a supercurrent flows on the surface (or just below the surface) of a wire. So, for example, if you had a circular cross-section 'wire' of pure niobium that was one inch in diameter, all the current would flow in a thin layer near the surface (when a current is flowing and it's not just a static situation). But if all the valence electrons, throughout the wire, move to the near surface conducting layer, at zero K, and participate in the supercurrent, it seems like there would be a strong charge imbalance within the one inch diameter wire. I suppose if the wire diameter is twice the Cooper pair coherence length, then the electrons in the wire's interior space could participate in the supercurrent without actually being in the near surface layer.

I found a great article about the pseudogap phenomena in high Tc superconductors, which I'll be reading shortly: http://www.science20.com/news_articles/what_pseudogap-77490 I briefly looked at the beginning of it, but will read the rest when I print it out at home.

Could the fact that less than 20% of free charge carriers participate in the supercurrent, in high Tc superconductors, result from the fact that only the copper-oxygen layers are actually involved in conduction?