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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:. - PowerPoint PPT Presentation
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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