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. The Saha Equation
Atoms within a star are ionised via interaction with photons
We have a dynamic equilibrium between photons and atoms on one hand and electrons and ions on the other
Considering the case of hydrogen: H + g « e- + p

20. The Saha Equation
Thermodynamic equilibrium is reached when the chemical potentials on both sides of the equation are equal
I.e, changes in numbers of particles doesn’t affect the energy, hence: m(H) + m(g) = m(e-) + m(p)

21. The Saha Equation Chemical potential of a photon: m(g) = 0
Also have to allow the hydrogen atom to be in any electronic quantum state, q, with energy: Eq = -13.6/q2 eV
Then: m(Hq) = m(e-) + m(p) (1)

22. The Saha Equation
Assuming the density is low and energies are non-relativistic:
See Workshop 3, Question 1
Evaluate the chemical potentials in terms of the quantum concentrations using functions derived in Lecture 5:

23. The Saha Equation For protons: For atoms: For electrons:
(Note nQ depends on mass and is almost identical for protons and hydrogen atoms)

24. The Saha Equation Also, gse = gsp = 2 and gsH = gq gse gsp with gq =q2
Note that the total energy of a hydrogen atom in state q is:
Also, gse = gsp = 2 and gsH = gq gse gsp with gq =q2

25. The Saha Equation
Combining these relationships with the condition for equilibrium (equation (1)), we obtain:

26. Consequences Degree of ionisation
Sum over all q levels to obtain the ratio of protons to all neutral states of H

27. Degree of Ionisation We may re-write this in the form:
The summation is truncated at a value of q such that the spatial extent of the atom is comparable to the separation of the atoms
In practice, the summation ~1

28. Degree of Ionisation
The ratio of ionised to neutral hydrogen (or indeed, any atom) can now be written as:

29. Degree of Ionisation Degree of ionisation in the sun:
Average density, r~1.4x103 kg/m-3
Typical temperature T~ 6x106 K
nQe ~ 1021T3/2
Assume electron density ~ proton density

30. Degree of Ionisation
Denote the fraction of hydrogen ionised as x(H+). Then, we can write:
ne = n(H+) = x(H+).r/mH
n(H) = (1-x(H+)).r/mH
We can now re-write (2) as:

31. Degree of Ionisation
Substituting the values for the sun and for hydrogen, we obtain x(H+) ~ 95%
I.e., the interior of the sun is almost completely ionised
For further discussion, see Phillips, ch. 2 secn 2.5

32. Balmer Absorption
To find the degree of Balmer absorption in a stellar atmosphere, we require: n(H(2))/(n(H)+n(H+)) (3)
Saha gives us n(H+)/n(H)
Boltzmann gives us n(H(2))/n(H(1))
Assume n(H(1))~ n(H)

33. Balmer Absorption
From Boltzmann
We can rewrite (3) as:
From Saha

34. Balmer Absorption
Hence:
Typically, ne ~ 1019 m-3

35. Balmer Absorption A B F O G K M

36. Reminder
Assignment 1 available today on module website

37. Next Week Private Study Week - Suggestions Assessment Worksheet
Review Lectures 1-3
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