Helium-4 is famous for its ability to form a Bose-Einstein condensate (BEC), because the atom is, in its ground state, a boson. This happens because the number of protons equals the number of neutrons and they're both even. The next atom for which this should be true would be Beryllium-8, but it has an absurdly short half-life. That leaves us with carbon-12 being the next lightest atom that should be a boson in its ground state.
Before today, I'd only ever heard of of BECs formed by helium and photons. Wikipedia's Bose-Einstein condensate article also lists rubidium-87 and sodium-23. That makes it look like the requirement for the individual atoms to be bosons may not actually be a requirement, and may even be a disadvantage since it lowers the cross-section for laser cooling.
Regardless, under what conditions should carbon-12 form a BEC? The only mention of carbon-12 BECs I find in the literature, that isn't about atomic physics, is the statement, "While a BEC of carbon-12 is not practical[…]," in the context of using carbon-12 in defining Avogadro's number in Appendix B of this paper about using properties of BECs to measure the mass of atoms. This statement has no other qualifications, so I assume that the difficulties are considered to be textbook level knowledge in the field.
The critical temperature formula in Wikipedia's article
$$
T_c = \frac{2\pi\hbar^2}{m k_B} \left(\frac{n}{\zeta(3/2)}\right)^{2/3}
$$
is derived under the assumption that the particles are in a non-interacting gas. Is that the problem that makes carbon-12 difficult to get into a BEC? That is, is the problem that to get carbon-12 to not strongly interact $n$ must be very low, making $T_c$ impractically low for an atom that (I assume) can't be easily laser cooled? I would assume that rubidium-87 and sodium-23 have similar problems, so is it something else?
Best Answer
BECs nowadays have been made of Rb, Na (these two were the first ones in 1995, that won the Nobel prize), He, Cs, Li, Yb, Sr, Er, Dy, Ca, and Cr. From what I hear they are trying to condense Fe and Ti. And these are just form what I can list from memory from when I finished my PhD, they may have done more by now.
The boring answer to you good question is that Bose condensing elements is usually not limited by the properties of the elements itself (e.g. $^{39}$K is naturally unstable but can be made stable with something called a Feshbach resonance to allow it to Bose condense), but to the lack of commercially available equipment to perform magnetic trapping and laser cooling, two of the basic techniques that lead to Bose condensation.
I cannot find a decent energy diagram of Carbon-12 online, but this quora article claims that the total magnetic moment of Carbon would be zero, and hence magnetically untrappable. If you told the wavelength of the transitions in C I could tell you whether laser exist to address them.