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Thermodynamics and Statistical Mechanics

Thermodynamics and Statistical Mechanics. Fermi-Dirac Statistics. Free Electrons in Metals.

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Thermodynamics and Statistical Mechanics

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  1. Thermodynamics and Statistical Mechanics Fermi-Dirac Statistics Thermo & Stat Mech - Spring 2006 Class 22

  2. Free Electrons in Metals • A model based on Fermi-Dirac statistics explains why metals do not have a much larger heat capacity than insulators. At one time it was expected that the “free electron gas” would provide a large addition to the lattice heat capacity, but it does not. Thermo & Stat Mech - Spring 2006 Class 22

  3. Fermi Function Thermo & Stat Mech - Spring 2006 Class 22

  4. Properties of Fermi Function Thermo & Stat Mech - Spring 2006 Class 22

  5. Properties of Fermi Function Thermo & Stat Mech - Spring 2006 Class 22

  6. Properties of Fermi Function m = 0.1 eV T1 = 1K T2 = 200K T3 = 400K Thermo & Stat Mech - Spring 2006 Class 22

  7. Free Electron Gas Thermo & Stat Mech - Spring 2006 Class 22

  8. Energy of Electrons Thermo & Stat Mech - Spring 2006 Class 22

  9. Fermi Energy eF Energy of highest energy state occupied at T = 0 K Thermo & Stat Mech - Spring 2006 Class 22

  10. Electrons’ Energy at T = 0 K Thermo & Stat Mech - Spring 2006 Class 22

  11. Electrons’ Energy at T > 0 K Thermo & Stat Mech - Spring 2006 Class 22

  12. Heat Capacity of Electrons Thermo & Stat Mech - Spring 2006 Class 22

  13. Heat Capacity of Metals • At normal temperatures, the heat capacity due to the electrons is much less than the heat capacity of the lattice, since kT << eF. • At low temperatures, the lattice heat capacity gets very small, and the contribution of the electrons becomes significant. Thermo & Stat Mech - Spring 2006 Class 22

  14. At Low Temperatures Thermo & Stat Mech - Spring 2006 Class 22

  15. Better Value for Electron’s Energy To calculate the energy of the electrons correctly, it is necessary to carry out the integral below. In order to do that, a value is needed for the chemical potential, m. Thermo & Stat Mech - Spring 2006 Class 22

  16. Approximate Method Thermo & Stat Mech - Spring 2006 Class 22

  17. Approximate Method Thermo & Stat Mech - Spring 2006 Class 22

  18. Graph of Thermo & Stat Mech - Spring 2006 Class 22

  19. Expansion Thermo & Stat Mech - Spring 2006 Class 22

  20. Evaluate each term Thermo & Stat Mech - Spring 2006 Class 22

  21. Evaluate each term Thermo & Stat Mech - Spring 2006 Class 22

  22. Apply to Chemical Potential Thermo & Stat Mech - Spring 2006 Class 22

  23. Chemical Potential Thermo & Stat Mech - Spring 2006 Class 22

  24. Chemical Potential Thermo & Stat Mech - Spring 2006 Class 22

  25. Finally, the Chemical Potential! Thermo & Stat Mech - Spring 2006 Class 22

  26. Electron Energy Thermo & Stat Mech - Spring 2006 Class 22

  27. Approximate again! Thermo & Stat Mech - Spring 2006 Class 22

  28. Still more! Thermo & Stat Mech - Spring 2006 Class 22

  29. And more! Thermo & Stat Mech - Spring 2006 Class 22

  30. Finally!! Thermo & Stat Mech - Spring 2006 Class 22

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