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In 1995 Wolfgang Ketterle at MIT produced a Bose-Einstein Condensate in a gas of sodium-23 atoms, but sodium-23 doesn't have an integer spin. How does this work?

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The protons, neutrons, and electrons that make up any atom are all Fermions with spin $1/2$. You can't make a Bose-Einstein condensate out of just electrons, for instance. But a composite particle can have a net integer spin.

As an example, a helium-4 nucleus has two protons and two neutrons. The nucleons are arranged according to the nuclear shell model. Both the neutrons and protons are paired resulting in a net spin of zero. In a helium atom there are two additional electrons, which are also paired, so the whole neutral helium atom has a net spin of zero. A helium-4 atom is a composite boson.

Sodium-23 has nuclear spin of $3/2$, making it a fermion. There are 12 paired neutrons, 10 paired protons, and one leftover unpaired proton. The unpaired proton sits in a shell state which contributes the spin of $3/2$. But the Bose-Einstein condensate is formed by atomic sodium, not nuclear sodium. The 11 electrons in a neutral sodium atom contribute an unpaired spin of $1/2$ to the total. The full sodium atom is a composite boson, so it can form a Bose-Einstein condensate.

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    $\begingroup$ If the sodium atoms are all in the same state, and the electrons are part of the atoms, are the electrons all in the same state? Despite being fermions? Is it even valid to talk about the state of an individual electron in a condensate of atoms? $\endgroup$
    – user253751
    Apr 26 at 14:07
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    $\begingroup$ The composite system of the whole atom, can be thought of as a single quantum mechanical wavefunction. It doesn't really make sense to think about one electron in isolation. This is how it works for any composite boson. Another example is Cooper pairs in a superconductor. A Cooper pair is a bound state of two electrons. The pair acts like a single boson which can condense into the superconducting state. $\endgroup$
    – Paul T.
    Apr 26 at 14:34
  • $\begingroup$ despite being separated by significant distances, the electrons in a pair do not behave as single electrons? A composite can be something completely different than the sum of its parts? $\endgroup$
    – user253751
    Apr 26 at 14:41
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    $\begingroup$ More generally, a neutral atom of any isotope with an even number of neutrons will be a boson (since the number of protons plus the number of electrons is always even for a neutral atom.) $\endgroup$ Apr 26 at 19:20

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