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?