Title: Thermodynamics and Statistical Mechanics
1Thermodynamics and Statistical Mechanics
2Free 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.
3Fermi Function
4Properties of Fermi Function
5Properties of Fermi Function
6Properties of Fermi Function
m 0.1 eV T1 1K T2 200K T3 400K
7Free Electron Gas
8Energy of Electrons
9Fermi Energy
eF Energy of highest energy state occupied at T
0 K
10Electrons Energy at T 0 K
11Electrons Energy at T gt 0 K
12Heat Capacity of Electrons
13Heat 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.
14At Low Temperatures
15Better 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.
16Approximate Method
17Approximate Method
18 Graph of
19Expansion
20Evaluate each term
21Evaluate each term
22Apply to Chemical Potential
23Chemical Potential
24Chemical Potential
25Finally, the Chemical Potential!
26Electron Energy
27Approximate again!
28Still more!
29And more!
30Finally!!