Physics: Problems and Solutions

UVa SM:Bose-Einstein Condensation temperature

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(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


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