How to Determine Chemical Potential of 44 ^6Li Atoms in a 3D Harmonic Well?

[Mattofix]

Homework Statement

What is the chemical potential, \mu (in units of \hbar\omega) at T = 0 K of 44 ^{6}Li atoms with spin 1/2 contained in a 3D harmonic potential well (energy levels, \epsilon_{i}=(i+3/2)\hbar\omega with degeneracies g_{i}= 1/2(i+1)(i+2) )?

Homework Equations

The Attempt at a Solution

spin = 1/2 \Rightarrow Fermoins

t=0 \Rightarrow \mu=\epsilon_{f} (fermi energy)

therefore:

= 1/2

So i need 'i' to then use in \epsilon_{i}=(i+3/2)\hbar\omega to obtain \mu.

I know that \frac{n_{i}}{g{i}} = 1/2 and that g_{i}= 1/2(i+1)(i+2) so all i need is the value of n_{i}, which is the number of particles in each g_{i}.

Is this all correct? If how do i obatin n_{i}?

 

[Mattofix]

Anyone got any ideas?