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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?

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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.

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.

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  • $\begingroup$ Interesting. Thanks! Do you think an elementally pure diamond crystal (or graphene sheet) could enter a BEC state at low enough temperature? A supersolid? $\endgroup$ Commented Sep 12, 2022 at 3:06
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    $\begingroup$ This will likely turn into a longer story. My answer is no, but I am not 100% sure. First of all, BEC is a non-interacting phenomenon, which has been observed to withstand weak interactions but it's not fully understood yet. To form a solid you do need interactions. Also, a solid is usually the "true" ground state of any system for low enough temperature. That is because you need 3 atoms to make a molecule (the 3rd to conserve momentum and energy), and the three-body recombination goes as density²interaction^4. So the denser the sample, the faster the molecule creation and the sooner it turns $\endgroup$ Commented Sep 12, 2022 at 3:25
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    $\begingroup$ into a solid. Hence, experiments creating BECs need to use a very dilute (low density) gas to ensure its lifetime against recombination loss is long enough to do experiments. Bottom line I am not sure a solid could ever become a BEC. I don't know much about supersolids. From what I understand they have some spatial ordering while retaining superfluid flow, so like matter that can flow through itself, and with zero resistance? I don't even know if it's an equilibrium state $\endgroup$ Commented Sep 12, 2022 at 3:26

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