# Why is Rubidium-87 used for Bose-Einstein condensates?

Rubidium-87 was the first and the most popular atom for making Bose–Einstein condensates in dilute atomic gases. Even though rubidium-85 is more abundant, rubidium-87 has a positive scattering length, which means it is mutually repulsive, at low temperatures. This prevents a collapse of all but the smallest condensates. It is also easy to evaporatively cool, with a consistent strong mutual scattering.

I am confused by the above paragraph, from Wikipedia's article "Isotopes of rubidium".

It says that Rubidium-87 atoms are mutually repulsive, which prevents a collapse......

But isn't a 'collapse' exactly what you want from a BEC? You DON'T want want them to be repulsive, but to come together in one single quantum state, unlike a fermion?

For a BEC, you want atoms to be in the same quantum state, not necessarily at the same position.

For a BEC, the temperature is low enough so that the de Broglie wavelength $$\lambda_{\mathrm{dB}} \propto 1/\sqrt{T}$$ is larger than the interatomic spacing $$\propto n^{-1/3}$$, $$n$$ being the density. This means that the wave nature of the atoms is large enough for it to be felt by other atoms, in other words atoms "see" each other even without exactly sitting on top of each other. This is just to further justify the claim that you don't need atoms at the same position. Actually, if you had a perfect box potential of side $$L$$, and you reached BEC, then the atoms will macroscopically occupy the ground state $$|\Psi|^2 \propto \sin^2(x/L)$$ which is very much extended. If you let $$L\rightarrow \infty$$, the atomic distribution becomes flat. So, again, very much atoms not at the same positions.

Ok, so now interactions and collapse.

First of all, BEC is a non-interacting effect. It is not driven by a competition of interaction terms, but solely by Bose-Einstein statistics. It is experimentally interesting that BEC seems to exist also in interacting systems, though there is no general theoretical proof. By BEC in an interacting system I mean macroscopic occupation of the ground state + Off-Diagonal Long-Range Order (ODLRO) — so not all superfluids are BECs. Let me also point out that you need interactions to reach a BEC as you need to reach thermal equilibrium.

The interaction strength among weakly interacting Bose-condensed bosons is quantified by a $$g n$$ term in the Hamiltonian, where $$g$$ is $$4\pi\hbar^2 a/m$$ (Gross-Pitaevski equation). You can make this interaction attractive with $$a<0$$ and repulsive with $$a>0$$, where $$a$$ is the scattering length and it is given by $$a(B) = a_0 f(B)$$, where $$a_0$$ is the background scattering length in the presence on no external magnetic field $$B$$ ($$f$$ is some function).

The pressure of a weakly interacting Bose-condensed gas is: $$P = -\frac{\partial E}{\partial V} = \frac{1}{2}gn^2.$$

Because $$n^2$$ is always positive, the condition for stability (i.e. not to collapse) is $$P>0$$ and hence $$g>0 \Rightarrow a>0$$ i.e. a repulsive system. With a positive pressure, the gas expands until it hits a wall (e.g. the confining potential). But if $$P<0$$ then the system is intrinsically unstable and collapses.

Rb-87 is "easy" because its background scattering length is positive and therefore trivially allows for a stable BEC. K-39, on the other hand, has a negative background scattering length so its "BEC" would collapse (and eventually explode). But its scattering length can be made repulsive by the use of a Feshbach resonance (applying a field $$B$$ to change $$a$$) so that it can undergo BEC.

## Interactions

Regarding attractive vs. repulsive interactions. Your original intuition that you would want attractive interactions for a BEC is understandable. You want the atoms to be very cold and densely formed so that they bose condense. Surely attractive interactions would bring the atoms closer? This chain of reasoning is incorrect. As SuperCiocia points out, the BEC transition is a non-interacting effect. If you had a non-interacting gas of bosons which was cold and dense enough it would form a BEC. Of course, again as SuperCiocia points out, in practice interactions are required for thermalization but that is a detail from the perspective of the phase transition.

So that said, we should understand that BEC doesn't happen because "the atoms are all attracted to eachother in a clump"*. Once we rid ourselves of this misconception we can then ask how does the introduction of interactions into the problem change things? The answer is short.

Strong attractive interactions cause the atoms to violently fly towards eachother resulting in inelastic collisions in which atoms can gain so much energy that they are ejected from the trap which is holding the atoms. The dynamics in light of attractive interactions is that you will just see rapid atom loss and no condensation. This is the collapse of the condensate.

What about repulsive interactions? A BEC can survive in spite of repulsive interactions. The atoms will be a little further apart than they would be without interactions but much of the major physics is unchanged.

So you should think like this: 1) theoretically our starting point is always a non-interacting BEC. Then we add in interactions. 2) If the interactions are attractive we get collapse. 3) If the interactions are repulsive things are changed/renormalized slightly but much of the essential physics is unchanged.

## Why Rb-87?

Rb-87 was more attractive for initial BEC than Rb-85 because Rb-87 supports an accessible cycling transition which could be used with early magneto optical trap and optical molasses technologies for laser cooling. Rb-85 does not support such a transition so more sophisticated laser cooling stages would have been necessary for the initial stages of cooling towards BEC. That is to say, Rb-87 probably wasn't specially chosen for its natural abundance

Why Rb among all other atoms? I can't speak much to this since I've mostly worked with Rb so far but I can point out that BECs of Na (for which the nobel prize was shared) and Li were formed shortly after the Rb BEC so I don't think we should infer something especially unique about Rb compared to other elements that it was the first to be condensed.

All of that said, I think your main question was really a confusion about interactions and BEC collapse which I think has been answered by now. It turns out that Rb has the right sign for interactions so that was helpful for it historic condensation.

*The question of why BEC does happen I will leave for you to research on your own or ask another question. The short story is that it is a thermodynamic transition that depends essentially on the fact that you have bosonic statistics and the density of states for the system.

• Interestingly, there is a phase of matter well characterised by "the atoms are all attracted to eachother in a clump" at that temperature - it's called a solid. (In other words, most of the work in creating a BEC is to keep the cloud at a low enough density to stop it from condensing; repulsive interactions are a large part of that.) – Emilio Pisanty Aug 2 at 20:35