# Bose-Einstein condensate in a particular system

Gold Member

## 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?

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 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!