Changes: UVa SM:Bose-Einstein Condensation temperature

Revision as of 02:18, 26 February 2007

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 $T_{B}EC$ 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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 ==Problem==
-(a)Find the Bose-Einstein condensation temperature <math>T_BEC</math> for a large number, <math>N</math>, of non-interacting atoms of mass <math>M</math> confined to a volume <math>V</math>. 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 <math>T \le T_{BEC}</math> <math>T_BEC</math>? What can be said about the value of <math>T_BEC</math> by dimensional analysis alone?
+(a)Find the Bose-Einstein condensation temperature <math>T_{BEC}</math> for a large number, <math>N</math>, of non-interacting atoms of mass <math>M</math> confined to a volume <math>V</math>. 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 <math>T \le T_{BEC}</math> <math>T_{BEC}</math>? What can be said about the value of <math>T_BEC</math> by dimensional analysis alone?
 (b)How can one create, in practice, such a BE condensate? (1 point)
-(c)Estimate the number density <math>N/V</math> necessary to obtain <math>T_BEC = 1</math> microKelvin if the atoms are <math>{}^{23}Na</math> (1 point).
+(c)Estimate the number density <math>N/V</math> necessary to obtain <math>T_{BEC} = 1</math> microKelvin if the atoms are <math>{}^{23}Na</math> (1 point).

Changes: UVa SM:Bose-Einstein Condensation temperature

Revision as of 02:18, 26 February 2007

Problem

Solution