Bose-Einstein condensate in a particular system

[fluidistic]

Homework Statement

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?

Homework Equations

Several.

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 believe 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!

 

[fluidistic]

I solved my problem. I had to check out the z limits only via the values for mu and forget about temperature for a moment.