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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.

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Classical and Quantum Gases

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  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. 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

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