# Bose-Einstein condensation of Helium

So I'm recalling this derivation of Bose-Einstein condensation in helium in some Thermo-stat mech text book. I thought it was in Reif, but I couldn't find it. It calculates the condensation temperature in He, (with pure QM, no He-He interaction) to be something like $$1.4 ~K$$. (the number is not important, I just want to remember the calculation.) Does anyone know the reference I'm talking about, or another similar calculation?

• You can find it in any book, those are especially written on the BE condensation (like the famous book by Pitaevskii & Stringari).
– SCh
Jan 19 at 2:30

There are numerous textbooks alongwith a lot of reliable online resources, where you can find the derivation. The famous some of them, which I've read includes,

1. Bose-Einstein Condensation by Lev. P. Pitaevskii & Sandro Stringari
2. Bose-Einstein Condensation by A. Griffin
3. Bose–Einstein Condensation in Dilute Gases by C. J. Pethick & H. Smith
4. Bose-Einstein Condensates: Theory, Characteristics & Current Research, editted by Paige W. Matthews
5. Universal Themes of Bose-Einstein Condensation by Nick P Proukakis, David W Snoke & Peter B Littlewood

But sometimes, it is better to have a overall idea about the derivation and then going through the textbooks (if further needed). So I'm giving here a short outline of the proof:

Let the density of states of the system be $$D(\epsilon)$$. So in D-dimensions (volume $$V=L^D$$) for infinite square well potential, $$D(\epsilon)~d\epsilon = \left(2S+1\right)\dfrac{L^D}{\Gamma\left(\frac{D}{2}\right)} \left(\dfrac{2m\pi}{\hbar^2}\right)^{D/2}~\epsilon^{\frac{D}{2}-1}~d\epsilon$$

and occupation number density, $$n(\epsilon) = \dfrac{1}{exp\left[\beta\left(\epsilon-\mu\right)\right]-1}$$ where $$\beta = \dfrac{1}{k_B T}$$

$$\therefore$$ Total number of particles in the system (ideal Bose gas), $$N = \displaystyle\int_0^\infty n(\epsilon)D(\epsilon)~d\epsilon = \left(2S+1\right)\dfrac{L^D}{\Gamma\left(\frac{D}{2}\right)} \left(\dfrac{2m\pi}{\hbar^2}\right)^{D/2}~\displaystyle\int_0^\infty \dfrac{\epsilon^{\frac{D}{2}-1}}{exp\left[\beta\left(\epsilon-\mu\right)\right]-1}~d\epsilon$$

In unit-less parameter, $$x=\beta\epsilon$$ and $$\bar\mu = \mu\beta$$, the integration becomes,

$$\Longrightarrow N = \left(2S+1\right)\dfrac{L^D}{\Gamma\left(\frac{D}{2}\right)} \left(\dfrac{2m\pi}{\hbar^2}\right)^{D/2}~\left(k_B T\right)^{\frac{D}{2}}~\displaystyle\int_0^\infty \dfrac{x^{\frac{D}{2}-1}}{e^{x-\bar{\mu}}-1}~d\epsilon$$

Let $$T$$ is being lowered, but $$N$$ is fixed. So the integral must increeased. Hence $$\bar\mu$$ must increase upto maximum valuie of zero$$(0)$$ (as $$\mu \leq \epsilon_{min}$$).

So it is possible that at a finite temperature $$(T_C)$$, we will arrive exactly at $$\bar\mu=0$$. Then, $$N = \left(2S+1\right)\dfrac{L^D}{\Gamma\left(\frac{D}{2}\right)} \left(\dfrac{2m\pi}{\hbar^2}\right)^{D/2}~\left(k_B T\right)^{\frac{D}{2}}~\displaystyle\int_0^\infty \dfrac{x^{\frac{D}{2}-1}}{e^{x}-1}~d\epsilon = \left(2S+1\right)L^D \left(\dfrac{2m\pi k_B T}{\hbar^2}\right)^{D/2}~\zeta\left(\dfrac{D}{2}\right)$$

It will converge, provided $$D \gt 2$$. Here $$T_C$$ is the condensate temperature.

At $$T=T_C$$, $$\bar\mu=0$$ is reached and at $$T \lt T_C$$, $$\bar\mu$$ cannot change. So the number of particles coming from the integral will be smaller than the actual number $$N$$.

$$\Longrightarrow$$ The remaining particles will settle in ground state with $$\epsilon=0$$ (this is Bose-Einstein condensate), let this number be $$N_0$$.

$$\therefore N = N_0 + \left(2S+1\right)\dfrac{L^D}{\Gamma\left(\frac{D}{2}\right)} \left(\dfrac{2m\pi k_B T}{\hbar^2}\right)^{D/2}~\zeta\left(\dfrac{D}{2}\right) = N_0 + N~\left(\dfrac{T}{T_C}\right)^{\frac{D}{2}}$$ for $$T \lt T_C$$

For $$3D$$, $$N_0 = N~\left[1-\left(\dfrac{T}{T_C}\right)^{\frac{3}{2}}\right]$$ $$\Longrightarrow \dfrac{N_0}{N} = 1-\left(\dfrac{T}{T_C}\right)^{\frac{3}{2}}$$

This is expression is so-called condensate fraction.

• This is great! Thank you. This will sound like I'm a bit of a nutcase, but I want to use this as a framework to calculate B-E condensation of gravitons in the universe. (I know many problems/ unknowns) Jan 19 at 17:53