UVa SM:Bose-Einstein Condensation temperature

Problem

(a)Find the Bose-Einstein condensation temperature $T_{BEC}$ for a large number, $N$, of non-interacting atoms of mass $M$ confined to a volume $V$. Assume the atoms have spin 0, or have integer spin but the spin degeneracy is lifted by an applied magnetic field. A derivation of the full result is required (8 points), but you can get partial credit by just answering such questions as: What is the momentum of an atom in the condensate? What is the value of the chemical potential for $T \le T_{BEC}$ $T_{BEC}$? What can be said about the value of $T_{BEC}$ by dimensional analysis alone?

(b)How can one create, in practice, such a BE condensate? (1 point)

(c)Estimate the number density $N/V$ necessary to obtain $T_{BEC} = 1$ microKelvin if the atoms are ${}^{23}Na$ (1 point).

Useful integral: $\int\limits_0^\infty {dx{{\sqrt x } \over {e^x - 1}}} = {{\sqrt \pi } \over 2}\zeta \left( {3/2} \right) \cong 2.3152$