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I've heard mentioned in various classes that neutron stars, like superconductors, are described by BCS theory. I know that in superconductors a key element in forming cooper pairs is a net attractive force between the electrons which would normally repel one another. That attractive force is accounted for via lattice vibrations (phonons) created and "absorbed" by electrons.

So my question is: what provides the attractive force between neutrons? just gravity? If it is true that neutron stars follow BCS theory, by what means was someone able to verify that?

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I'm afraid the actual situation is much more complicated than you've been told. For one thing, the superconductivity does not occur between neutrons, but between quarks themselves. The topic of high density QCD is a very cool interplay of condensed matter and high energy physics, and a very nice review is available by Frank Wilczek. However, that article does need some background in QCD and superconductivity simultaneously to appreciate.

A shortened version might go something like this:

  1. Inspiration: free fermions are incredibly unstable to superconductivity, in that any attractive interaction will cause it (in fact, there's an old theorem by Pierls (?) that almost all interactions (even repulsive ones) will cause superconductivity if you cool far enough). In QCD, quarks naturally attract already! So at sufficiently high density and low temperature, we can imagine that QCD can cause a strong attractive instability to a Fermi gas of quarks.
  2. Complications: 6 flavours, chiralities, masses of quarks are different, etc.
  3. Simplification by complication: realise that the normal state of QCD (i.e. 3- and 2-quark combinations) is just that: one possible state. Other phases of QCD exist, and we can study the phase boundaries and so forth even if we can't compute things exactly (universality saves the day!). We find that at really high densities, quarks pair up to give a background diquark condensate, through which single quarks move, and through the Anderson-Higgs mechanism gains a large mass by eating some Goldstone modes. All gluons become gapped (again, Anderson-Higgs), apart from one which mixes with the photon. The symmetries of this solution is actually the same as that of normal matter --- replace baryons with quarks (+ their diquark condensed background) and mesons with diquarks; this suggests that they are really the same phase in theory. In practise, getting from one to another requires some other phases in the middle, which are more complicated and arise due to the quark masses and the number of flavours, etc.
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To be fair, what you are describing here is color superconductivity, which only takes place at the core of the neutron star (but even that is not entirely certain, because the critical density for color superconductivity is not known). In the outer shell of the neutron star you do have ordinary superdconductivity of the electrons and protons and a superfluid of the neutrons, as far as I understand. –  Olaf Jun 15 '11 at 14:53
    
@MisterO: to be sure. I don't have any references to hand to suggest what the outer layers might be however; in those cases though, the mediating forces are sure to be much more complex (mesons?) and I wouldn't like to make any conjectures about whether one is below the transition temperature for those... –  genneth Jun 15 '11 at 15:37

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