1
$\begingroup$

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?

$\endgroup$
1
  • $\begingroup$ You can find it in any book, those are especially written on the BE condensation (like the famous book by Pitaevskii & Stringari). $\endgroup$
    – SCh
    Jan 19 at 2:30

1 Answer 1

2
$\begingroup$

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.

$\endgroup$
1
  • 1
    $\begingroup$ 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) $\endgroup$
    – George H.
    Jan 19 at 17:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.