1. 18.3 Bose–Einstein Condensation • A gas of non-interacting particles (atoms & molecules) of relatively large mass. • The particles are assumed to comprise an ideal B-E gas. • Bose – Einstein Condensation: phase transition • B – E distribution:

2. First Goal: Analyzing how the chemical potential μ varies with temperature T. • Choosing the ground state energy to be ZERO! At T = 0 all N Bosons will be in the ground state. μmust be zero at T = 0 μ is slightly less than zero at non zero, low temperature.

3. At high temperature, in the classical limit of a dilute gas, M – B distribution applies: In chapter 14: Thus

4. Example: one kilomole of 4He at STP = -12.43 The average energy of an ideal monatomic gas atom is Confirming the validity of the dilute gas assumption.

5. From chapter 12: There is a significant flow in the above equation (discussion … )

6. The problem can be solved by assuming: At T very close to zero, for N large

7. Using The Bose temperature TB is the temperature above which all bosons are in excited states. i.e. For

9. Variation with temperature of μ/kTB for a boson gas.

10. 18.4 Properties of a Boson Gas Bosons in the ground state do not contribute to the internal energy and the heat capacity. For Below

11. Assume each Boson has the energy kT More exact result:

14. 18.5 Application to Liquid Helium • Phase diagram

15. Helium phase diagram II

16. 18.14 In a Bose-Einstein condensation experiment, 107 rubidium-87 atoms were cooled down to a temperature of 200 nK. The atoms were confined to a volume of approximately 10-15 m3. (a) Calculate the Bose temperature

17. (b) Determine how many actoms were in the ground state at 200 nK. • (c) calculate the ratio of kT/ε0, where T = 200 nK and where the ground state energy ε0 is given by 3h2/(8mV2/3)

18. 18.6) assume that the universe is spherical cavity with radius 1026 m and temperature 2.7K. How many thermally excited photons are there in the universe? • Solution: equation 18.16 or 18.36 (?)