1. Classical and Quantum Gases
Fundamental Ideas
Density of States
Internal Energy
Fermi-Dirac and Bose-Einstein Statistics
Chemical potential
Quantum concentration

2. 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:
gs = number of polarisations
2 for protons, neutrons, electrons and photons

3. Internal Energy The energy of a particle with momentum p is given by:
Hence the total energy is:
No. of quantum states in p to p +dp
Average no. of particles in state with energy Ep

4. Total Number of Particles
No. of quantum states in p to p +dp
Average no. of particles in state with energy Ep

5. Fermi-Dirac Statistics
For fermions, no more than one particle can occupy a given quantum state
Pauli exclusion principle
Hence:

6. Bose-Einstein Statistics
For Bosons, any number of particles can occupy a given quantum state
Hence:

7. F-D vs. B-E Statistics

8. The Maxwellian Limit
Note that Fermi-Dirac and Bose-Einstein statistics coincide for large E/kT and small occupancy
Maxwellian limit

9. Ideal Classical Gases
Classical Þ occupancy of any one quantum state is small
I.e., Maxwellian
Equation of State:
Valid for both non- and ultra-relativistic gases

10. Ideal Classical Gases Recall: Non-relativistic: Ultra-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

11. Ideal Classical Gases
Total number of particles N in a volume V is given by:

12. Ideal Classical Gases
Rearranging, we obtain an expression for m, the chemical potential

13. Ideal Classical Gases Interpretation of m
From statistical mechanics, the change of energy of a system brought about by a change in the number of particles is:

14. Ideal Classical Gases Interpretation of nQ (non-relativistic)
Consider the de Broglie Wavelength
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 l and the gas is classical rather than quantum

15. Ideal Classical Gases
A similar calculation is possible for a gas of ultra-relativistic particles:

16. 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

17. Quantum Gases Equations of State: (See Physics of Stars secn 2.2)
Non-relativistic:
Ultra-relativistic:

18. 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

19. Reminder
Assignment 1 available today on unit website

20. Next Lecture The Saha Equation Derivation
Consequences for ionisation and absorption

21. 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