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

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Presentation transcript:

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

Thermo & Stat Mech - Spring 2006 Class 222 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 223 Fermi Function

Thermo & Stat Mech - Spring 2006 Class 224 Properties of Fermi Function

Thermo & Stat Mech - Spring 2006 Class 225 Properties of Fermi Function

Thermo & Stat Mech - Spring 2006 Class 226 Properties of Fermi Function  = 0.1 eV T 1 = 1K T 2 = 200K T 3 = 400K

Thermo & Stat Mech - Spring 2006 Class 227 Free Electron Gas

Thermo & Stat Mech - Spring 2006 Class 228 Energy of Electrons

Thermo & Stat Mech - Spring 2006 Class 229 Fermi Energy  F Energy of highest energy state occupied at T = 0 K

Thermo & Stat Mech - Spring 2006 Class 2210 Electrons’ Energy at T = 0 K

Thermo & Stat Mech - Spring 2006 Class 2211 Electrons’ Energy at T > 0 K

Thermo & Stat Mech - Spring 2006 Class 2212 Heat Capacity of Electrons

Thermo & Stat Mech - Spring 2006 Class 2213 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 <<  F. At low temperatures, the lattice heat capacity gets very small, and the contribution of the electrons becomes significant.

Thermo & Stat Mech - Spring 2006 Class 2214 At Low Temperatures

Thermo & Stat Mech - Spring 2006 Class 2215 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, .

Thermo & Stat Mech - Spring 2006 Class 2216 Approximate Method

Thermo & Stat Mech - Spring 2006 Class 2217 Approximate Method

Thermo & Stat Mech - Spring 2006 Class 2218 Graph of

Thermo & Stat Mech - Spring 2006 Class 2219 Expansion

Thermo & Stat Mech - Spring 2006 Class 2220 Evaluate each term

Thermo & Stat Mech - Spring 2006 Class 2221 Evaluate each term

Thermo & Stat Mech - Spring 2006 Class 2222 Apply to Chemical Potential

Thermo & Stat Mech - Spring 2006 Class 2223 Chemical Potential

Thermo & Stat Mech - Spring 2006 Class 2224 Chemical Potential

Thermo & Stat Mech - Spring 2006 Class 2225 Finally, the Chemical Potential!

Thermo & Stat Mech - Spring 2006 Class 2226 Electron Energy

Thermo & Stat Mech - Spring 2006 Class 2227 Approximate again!

Thermo & Stat Mech - Spring 2006 Class 2228 Still more!

Thermo & Stat Mech - Spring 2006 Class 2229 And more!

Thermo & Stat Mech - Spring 2006 Class 2230 Finally!!