1. Molecules and Solids (Cont.)
Chapter 43
Molecules and Solids (Cont.)
Dr. Jie Zou
PHY 1371

2. Outline Free electron theory of metals Band theory of solids
Dr. Jie Zou
PHY 1371

3. Free-electron theory of metals
Classical free-electron theory of electrical conduction in metals:
Model: Treats a metal as an electron gas and uses the kinetic theory of gases.
Predicts Ohm’s law
Difficulties: Does not predict the correct values of electrical and thermal conductivities.
Quantum-based free-electron theory of metals:
Model: The outer-shell electrons are free to move through the metal but are trapped within a three-dimensional box formed by the metal surfaces.
Dr. Jie Zou
PHY 1371

4. Fermi-Dirac distribution function
Quantum statistics: Required by the Pauli exclusion principle that each state of the system can be occupied by only two electrons (one with spin-up and the other with spin down).
The probability that a particular state having energy E is occupied by one of the electrons in a solid is given by
Function f (E) is called the Fermi-Dirac distribution function. EF is called the Fermi energy.
Quick quiz: Physical meaning of the two plots on the left.
Dr. Jie Zou
PHY 1371

5. Electron in a three-dimensional box
For a particle in a one-dimensional box of length L, the allowed value of energy is
For one electron in a solid cube of sides L and volume L3, the energy for such an electron is (see Problem 30)
For example, the ground state, nx = ny = nz =1 and E = 3ħ22/2meL2, can be occupied by two electrons corresponding to spin-up and spin-down (ms = +1/2 and ms = -1/2).
Dr. Jie Zou
PHY 1371

6. Density-of-states function
The number of allowed states per unit volume that have energies between E and E + dE is
Function g(E) is called the density-of-states function.
Dr. Jie Zou
PHY 1371

7. Quick quiz
The Fermi energy for silver is 5.48 eV. Near which of these energies are the energy levels closer together?
(a) 2 eV. (b) 6 eV. (c) The spacing of energy levels is the same near both energies.
Dr. Jie Zou
PHY 1371

8. Electron distribution function versus energy
Set N(E)dE = The number of electrons per unit volume that have energy between E and E + dE, where
N(E) is called the electron distribution function.
Dr. Jie Zou
PHY 1371

9. Problem 35
(a) Consider a system of electrons confined to a three-dimensional box. Find the ratio of the number of allowed energy levels at 8.50 eV to the number at 7.00 eV.
(b) Copper has a Fermi energy of 7.0 eV at 300 K. Calculate the ratio of the number of occupied levels at an energy of 8.50 eV to the number at the Fermi energy.
(c) Compare answers for (a) and (b).
Dr. Jie Zou
PHY 1371

10. Electron concentration
Set ne = The total number of electrons per unit volume. ne is called the electron concentration.
Find Fermi energy at T = 0 K from
Solve for
See Table 43.4 for the values of ne for different metals.
Dr. Jie Zou
PHY 1371

11. Band theory of solids
Dr. Jie Zou
PHY 1371

12. Energy bands of a sodium crystal
Blue represents energy bands occupied by the sodium electrons.
Gold represents energy bands that are empty.
Energy gaps or forbidden energies (white regions) between the allowed bands; electrons cannot occupy states that lie in these gaps.
Dr. Jie Zou
PHY 1371

13. Homework
Chapter 43, P. 1436, Problems: #33, 35.
Dr. Jie Zou
PHY 1371