Bose-Einstein condensate in a particular system

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fluidistic
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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 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!
 

Answers and Replies

  • #2
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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 
  • #3
fluidistic
Gold Member
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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.
 

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