## 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} $ 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 $