Chapter 43

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

2. Outline • Free electron theory of metals • Band theory of solids 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. 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. 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). 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. 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. 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. 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). 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. PHY 1371

11. Band theory of solids 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. PHY 1371

13. Homework • Chapter 43, P. 1436, Problems: #33, 35. PHY 1371