Download Classical and Quantum Gases

Classical and Quantum Gases

Fundamental Ideas
– Density of States
– Internal Energy
– Fermi-Dirac and Bose-Einstein Statistics
– Chemical potential
– Quantum concentration
Density of States

Derived by considering the gas particles
as wave-like and confined in a certain
volume, V.
– Density of states as a function of
momentum, g(p), between p and p + dp:
V
2
g  p dp  g s 3 4p dp
h
– gs = number of polarisations

2 for protons, neutrons, electrons and photons
Internal Energy

The energy of a particle with
momentum p is given by:
Ep  p c m c
2

2
2
2
4
Hence the total energy is:
 
E  0 E p f E p g  p dp

Average no. of particles
in state with energy Ep
No. of quantum states in
p to p +dp
Total Number of Particles
 
N  0 f E p g  p dp

Average no. of particles
in state with energy Ep
No. of quantum states in
p to p +dp
Fermi-Dirac Statistics

For fermions, no more than one particle
can occupy a given quantum state
– Pauli exclusion principle

Hence:
 
f Ep 
1
   1
exp
E p 
kT
Bose-Einstein Statistics
For Bosons, any number of particles can
occupy a given quantum state
 Hence:

 
f Ep 
1
   1
exp
E p 
kT
F-D vs. B-E Statistics
100
Fermi-Dirac
Bose-Einstein
1
0.01
0.1
1
E/kT
10
0.1
0.01
Occuapncy
10
0.001
0.0001
The Maxwellian Limit

Note that Fermi-Dirac and Bose-Einstein
statistics coincide for large E/kT and
small occupancy
– Maxwellian limit
 
  
f E p  exp 
E p 
kT
Ideal Classical Gases

Classical  occupancy of any one
quantum state is small
– I.e., Maxwellian

Equation of State:
N
P  kT
V

Valid for both non- and ultra-relativistic
gases
Ideal Classical Gases

Recall:
– Non-relativistic:


Pressure = 2/3 kinetic energy density
Hence average KE = 2/3 kT
– Ultra-relativistic


Pressure = 1/3 kinetic energy density
Hence average KE = 1/3 kT
Ideal Classical Gases

Total number of particles N in a volume
V is given by:
N 


0
 
exp
 E p
kT
V
g s 3 4 p 2dp
h
 
V
 N  g s 3 2mkT  exp
h
3
2
 mc 2
kT
Ideal Classical Gases

Rearranging, we obtain an expression
for , the chemical potential
g
n

s
Q 
2
  mc  kT ln

 n 
3
2
2

mkT


where nQ  
2

h


(the quantum concentration)
Ideal Classical Gases

Interpretation of 
– From statistical mechanics, the change of
energy of a system brought about by a
change in the number of particles is:
dE  dN
Ideal Classical Gases

Interpretation of nQ (non-relativistic)
– Consider the de Broglie Wavelength
h
h

 
 nQ
p mkT 
1
1
3
2
– Hence, since the average separation of
particles in a gas of density n is ~n-1/3
– If n << nQ , the average separation is
greater than  and the gas is classical
rather than quantum
Ideal Classical Gases

A similar calculation is possible for a gas
of ultra-relativistic particles:
 g s nQ 
  kT ln

 n 
 kT 
where nQ  8 

 hc 
3
Quantum Gases
Low concentration/high temperature
electron gases behave classically
 Quantum effects large for high electron
concentration/”low” temperature

– Electrons obey Fermi-Dirac statistics
– All states occupied up to an energy Ef , the
Fermi Energy with a momentum pf
– Described as a degenerate gas
Quantum Gases

Equations of State:
– (See Physics of Stars secn 2.2)
– Non-relativistic:
h
P 
5m
2
2
 3  3 53
 8  n
 
– Ultra-relativistic:
P 
hc  3 
2
3
n


4  8 
4
3
Quantum Gases

Note:
– Pressure rises more slowly with density for
an ultra-relativistic degenerate gas
compared to non-relativistic
– Consequences for the upper mass of
degenerate stellar cores and white dwarfs
Reminder

Assignment 1 available today on unit
website
Next Lecture

The Saha Equation
– Derivation
– Consequences for ionisation and
absorption
Next Week

Private Study Week - Suggestions
– Assessment Worksheet
– Review Lectures 1-5
– Photons in Stars (Phillips ch. 2 secn 2.3)


The Photon Gas
Radiation Pressure
– Reactions at High Temperatures (Phillips ch. 2 secn
2.6)


Pair Production
Photodisintegration of Nuclei