Solid State Physics 1

1. Solid State Physics1 Section 10-1

2. Topics • Structure of Solids • Free Electron Gas in Metals • Summary

3. Structure of Solids Face-centered cubic (fcc) Salts: LiF, KF, KCl, KI, NaCl Metals: Ag, Al, Au, Ca, Cu, Ni, Pb

4. Na+ Cl- Structure of Solids When two oppositely charged ions are brought together they experience an attractive electrostatic force. But, when their wave functions begin to overlap they experience the Coulomb repulsion of their electrons as well as the repulsion due to the exclusion principle

5. Na+ Cl- Structure of Solids The electric potential at a given ion is the sum of the electric potentials due to every other ion in the solid - + + - - + +

6. Na+ Cl- Structure of Solids Therefore, the net electrostatic potential energy of an ion is given by r For the 6 nearest neighbors of the chlorine ion the electrostatic potential energy is

7. Na+ Cl- Structure of Solids When the sum is extended over the whole solid, the factor of 6 is replaced by the factor a called the Madelung constant and the net electrostatic potential energy is given by r For NaCl a = 1.7476 and the equilibrium separation between the ions is r = 0.282 nm

8. Na+ Cl- Structure of Solids The potential energy associated with the repulsive force can be modeled using r So the total potential energy of an ion is given by

9. Na+ Cl- Structure of Solids If r0 is the equilibrium spacing, that is, the spacing for which the net force between the ions is zero, then we can write r

10. Na+ Cl- Structure of Solids Example 10-1: Compute r0, given that the density of NaCl is r = 2.16 g/cm3 r Each ion is associated with a cube of volume r03. The molar mass of NaCl is m = 58.4 g, therefore, and

11. Na+ Cl- Structure of Solids The dissociation energy is the energy required to break the bonds that hold the solid together. For NaCl it is measured to be 770 kJ/mol. The dissociation energy per ion-pair in eV is given by [(7.7 x 105J/mol)/(1.602 x 10-19 J/eV)]/ 6.022 x 1023 ion pairs/mol = 7.98 eV per ion pair r

12. Na+ Cl- Structure of Solids The cohesive energy is the energy per atom (or atomic pair). For NaCl we need 7.98 eV to remove a Na+ and a Cl- from the solid. To go from Cl- to Cl requires 3.62 eV and Na+ to Na releases 5.14 eV. Therefore, the cohesive energy is (7.98 + 3.62 – 5.14)/2 = 3.23 eV per atom. r

14. Structure of Solids Simple cubic (sc) Examples: CsCl, NH4Cl, Ba, Fe, K, Na

15. Structure of Solids Hexagonal close-packed (hcp) Examples: Be, Cd, Mg, Zn

16. Free Electron Gas in Metals • Solid metals are bonded by the metallic bond • One or two of the valence electrons from each atom are free to move throughout the solid • All atoms share all the electrons. A metal is a lattice of positive ions immersed in a gas of electrons. The binding between the electrons and the lattice is what holds the solid together

17. Free Electron Gas in Metals We first consider the physics of the electron gas. Since we are dealing with fermions, the electron gas obeys Fermi-Dirac statistics. Recall that the number of electrons with energy in the range E to E+dE is given by where, the density of states g(E) is given by

18. Free Electron Gas in Metals The electron has two spin states, so W = 2. If the electron’s speed << c, we can take its energy to be So the number of states in (E, E+dE) is given by Extra Credit: Derive this expression Due date: Friday, March 28

19. Free Electron Gas in Metals The total number of electrons N is given by the integral At T=0, the Fermi-Dirac distribution is equal to 1 if E < EF and 0 if E > EF. At T=0, all energy levels are filled up to the energy EF, called the Fermi energy.

20. Summary • The ions in solids form regular lattices • A metal is a lattice of positive ions immersed in a gas of electrons. All ions share all electrons • The attraction between the electrons and the lattice is called a metallic bond • At T = 0, all energy levels up to the Fermi energy are filled