# Bose-Einstein condensate in a particular system

1. Dec 11, 2014

### fluidistic

1. The problem statement, all variables and given/known data
Hi guys,
I'm having some troubles on a problem. There are N spinless particles with allowed energies $\vec p^2 /(2m)$ and $-\gamma$ where $\gamma >0$.
1)Find the number of particles with energy $-\gamma$ in function of T.
2)Which conditions must $\mu$ satisfy?
3)Is there a Bose-Einstein condensate?
4)What is the critical temperature of the condensate?

2. Relevant equations
Several.

3. The attempt at a solution
1)I got $n(-\gamma)=\frac{1}{e^{-\beta (\mu +\gamma ) -1}}$.
2)Since $n(-\gamma)$ must be non negative, I found out that $\mu \leq -\gamma$.
3)This is where the problems begin.
In order to check out if there's a condensate, I must investigate what happens with $<N>/V$.
I found out that $\frac{\langle N \rangle}{V}=\frac{1}{V}\cdot \frac{z}{e^{-\gamma \beta }-z} + \frac{g_{3/2}(z)}{\lambda _{\text{thermal}}}=\frac{1}{v}=\rho$.

I've got enormous troubles determining what values z takes when T tends to 0 and T tends to positive infinity. I know that $g_{3/2}(z)$ is bounded for $0\leq z \leq 1$.

z is the fugacity and is worth $e^{\beta \mu}$. Also, I am not 100% sure, but I believe that when T tends to positive infinity, mu tends to minus infinity. This complicates me a lot when I must figure out what value z takes in this case. But I believe that it's 0, I mean $z\to 0$.

Now when T tends to 0, I belive that $\mu \to -\gamma$. And so that z tends to 0, same case as when T tends to positive infinity... which cannot be.

I don't know where I go wrong. These limits are confusing me a lot.

Thanks for any light shedding!

2. Dec 16, 2014