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:

• 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.

μ must be zero at T = 0μ is slightly less than zero at non zero,

low temperature.

At high temperature, in the classical limit of a dilute gas, M – B distribution applies:

In chapter 14:

Thus

Example: one kilomole of 4He at STP

= -12.43The average energy of an ideal monatomic gas

atom is

Confirming the validity of the dilute gas assumption.

From chapter 12:

There is a significant flow in the above equation (discussion … )

The problem can be solved by assuming:

At T very close to zero,

for N large

Using

The Bose temperature TB is the temperature above which all bosons are in excited states.

i.e. For

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

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

Assume each Boson has the energy kT

More exact result:

18.5 Application to Liquid Helium

• Phase diagram

Helium phase diagram II

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

• (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)

• 18.6) 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 (?)

87

326

7

1066.1

10347.2

1002.2

N

VkT

VTN