Thermodynamics and Statistical Mechanics

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Thermodynamics and Statistical Mechanics

• Fermi-Dirac Statistics

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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.

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Fermi Function
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Properties of Fermi Function
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Properties of Fermi Function
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Properties of Fermi Function
m 0.1 eV T1 1K T2 200K T3 400K
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Free Electron Gas
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Energy of Electrons
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Fermi Energy
eF Energy of highest energy state occupied at T
0 K
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Electrons Energy at T 0 K
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Electrons Energy at T gt 0 K
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Heat Capacity of Electrons
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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 ltlt eF.
• At low temperatures, the lattice heat capacity
  gets very small, and the contribution of the
  electrons becomes significant.

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At Low Temperatures
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Better Value for Electrons 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.
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Approximate Method
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Approximate Method
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Graph of
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Expansion
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Evaluate each term
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Evaluate each term
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Apply to Chemical Potential
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Chemical Potential
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Chemical Potential
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Finally, the Chemical Potential!
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Electron Energy
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Approximate again!
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Still more!
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And more!
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Finally!!