Classical and Quantum Gases

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

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

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: • 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 electrons: • For protons: • For atoms: • (Note nQdepends on mass and is almost identical for protons and hydrogen atoms)

24. The Saha Equation • Note that the total energy of a hydrogen atom in state q is: • Also, gse = gsp = 2 andgsH = 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 obtainx(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. From Boltzmann From Saha Balmer Absorption • We can rewrite (3) as:

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