Lecture13-07.2-Bose_condensate-Fermi_gas

Lecture13-07.2-Bose_condensate-Fermi_gas - Last Time...

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1 Last Time Processes of an ideal gas: LECTURE 13 Reversible Isothermal Expansion Reversible Adiabatic, Isentropic Expansion Sudden (irreversible!) Expansion into Vacuum Bose Einstein Condensation N τ μ = - ( ) 0, 0 f T N ε = Today Chapter 7 (part 2) Fermi and Bose Gases LECTURE 13 Bose Einstein Condensation Excitations of a Bose Einstein Condensate Superfluidity Fermi Gases: Heat Capacity Application to Electrons in Metals
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2 Occupancies and Condensation The right hand side can only be so large -- to see how large, set μ = 0. Recognize this? Why is it the same as for fermions? (But μ will always be less than the ground state orbital.) Occupancies and Condensation Note that for low enough temperature, this can no longer be satisfied… Where can the particles go? The most the right hand side can be is: y ε τ y ε τ = d dy ε τ =
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3 Occupancies and Condensation Problem with integral at small ε ! 22 11 22 10 22 1 10 1 2 10 10 5 10 10 - - + × = + × Count occupation of the 2 first states: Count the same using integral: 2) Cannot use integral when terms in the sum are changing fast! 37 2.5 10 J ε - Δ = × 48 1.38 10 J μ - ≈ - × He 4 example, N=10 22 , 1cm 3 26 1.38 10 1 J mK τ - = × = 1 st state, f = 10 -22 2 nd state, f = 2x10 11 ε , in units Δε 10 15 Δε ° ( ε ) f ( ε ) ( ) ( ) 10 0 5 10 d f ε ε ε ε Δ × ° ° Ground state is missing! 1) ° (0) = 0 ( ) ( ) 1 1 f e ε μ τ ε τ ε μ - = - - Separate out the first orbital: He 2 at T = 1mK Occupancies and Condensation 37 2.5 10 J ε - Δ = × 48 1.38 10 J μ - ≈ - × 26 1.38 10 1 J mK τ - ≈ - × =
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4 Separate out the first orbital: ground state excited Occupancies and Condensation Einstein condensation temperature 2 2 3 2 2.61 E N M V π τ ± ² ³ ´ µ ° Excited atoms Atoms in the ground state Two Fluids All Normal Liquid Bose Einstein Condensation Occupancies and Condensation 2 2 3 2 2.61 E N M V π τ ± ² ³ ´ µ ° Note: only valid when large number is in ground state! Norm al fluid Superfluid Helium T E = 3K
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5 Liquid 4 He Condensation into liquid: 4.2 K Condensation into superfluid, predicted: 3 K measured: 2.17 K Liquid He behaves as a gas!
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