1. Copyright © 2012 Pearson Education Inc. PowerPoint ® Lectures for University Physics, Thirteenth Edition – Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Chapter 42 Molecules and Condensed Matter

2. Copyright © 2012 Pearson Education Inc. Goals for Chapter 42 To understand the bonds holding atoms together To see how rotation and vibration of molecules affect spectra To learn how and why atoms form crystalline structures To apply the energy-band concept to solids To develop a model for the physical properties of metals To learn how impurities affect semiconductors and to see applications for semiconductors To investigate superconductivity

3. Copyright © 2012 Pearson Education Inc. Ionic bonds There are several ways in which atoms can bind together to form more complex arrangements, each using a somewhat different type of electrostatic attraction. Two strong bond types are ionic bonds and covalent bonds (1-5 eV), while weaker bonds (0.5 eV or less) include van der Waals bonds and hydrogen bonds. An ionic bond is an interaction between oppositely charged ionized atoms, such as Na + and Cl ., with a binding energy of 4.2 eV. Figure 37.1 (right) shows a graph of the potential energy of two oppositely charged ions. Example 42.1: What is the electrostatic potential energy of a pair of Na + and Cl  ions separated by their equilibrium distance of 0.24 nm? The actual energy is  5.7 eV. Consider the Na + and Cl  ions as +e and –e charges.

4. Copyright © 2012 Pearson Education Inc. Covalent bonds In a covalent bond, the electrons are more shared among the atoms of the molecule. The wave functions of the outer shells are distorted and become more concentrated in certain places. Figure 42.2 (right) shows the hydrogen covalent bond (binding energy  4.8 eV, and Figure 42.3 (below) shows the methane molecule.

5. Copyright © 2012 Pearson Education Inc. Rotational energy levels Once two or more atoms are bound into a molecule, they can undergo various rotations and vibrations. Because of their small sizes, these various motions are quantized in certain allowed energy states. Rotation of a 2-atom molecule might be modeled as below. As we saw earlier, the relevant mass of both rotational and vibrational motions is the reduced mass, There is a nice analogy with classical rotation: For an isolated atom U = 0, so

6. Copyright © 2012 Pearson Education Inc. Rotational energy levels The angular solutions to the Schrödinger equation are the same as those for the hydrogen molecule (both have no angular dependence of U), hence as before Combining this with the “classical” energy dependence on L, we have quantized energies: Example 42.2: Carbon monoxide atoms are 0.1128 nm apart. The atomic masses are m C = 1.993 x 10 -26 kg, m O = 2.656 x 10 -26 kg. (rotational energy levels for diatomic molecule) (a) Find energies of the lowest three rotational energy levels of CO. (b) Find wavelength of photon emitted in transition from l = 2 to l = 1 rotational level.

7. Copyright © 2012 Pearson Education Inc. Vibrational energy levels In a similar manner, a diatomic molecule can vibrate, as in the classical analog below, two masses on a spring. Both atoms vibrate about their center of mass, so once again the relevant mass to use is the reduced mass. This system has the same energy levels as the quantum-mechanical harmonic oscillator, which was in Chapter 40, although we did not spend time on it then. The energy levels are given by Figure 42.7 shows some vibrational energy levels of a diatomic molecule. Combining both rotation and vibration, one has:

8. Copyright © 2012 Pearson Education Inc. Rotation and vibration combined Combining both rotation and vibration, one has: Figure 42.8 (right) shows an energy-level diagram for rotational and vibrational energy levels of a diatomic molecule. Figure 42.9 (below) shows a typical molecular band spectrum. Quantum-mechanical rules require  l = ±1 and  n = ±1 (plus for absorbing a photon, and minus for emitting a photon). The arrows in the figure show allowed transitions between from n = 2 levels.

9. Copyright © 2012 Pearson Education Inc. Complex Molecules More complex molecules like CO 2 have additional vibration modes as shown schematically at the right. Each of these motions follows its own quantization rules, with separations in energy less than 1 eV, so they produce infrared photons with wavelength longer than 1  m. This fact makes CO 2 a very effective “greenhouse” gas, that absorbs heat from the ground and traps it in the atmosphere. Venus’ atmosphere is nearly entirely CO 2, hence the planet’s surface temperature is near 800 K. Methane (CH 4 ) is an even more effective greenhouse gas.

10. Copyright © 2012 Pearson Education Inc. Crystal lattices The next simplest multi-atom structures to consider are crystalline materials. A crystal lattice is a repeating pattern of mathematical points. Figure 42.11 (below) shows some common types of lattices. These are “simple” because of their repeating structures, although actual crystals have imperfections that can strongly influence their bulk behavior. Ionic or covalent bonds can occur in crystals, which are called either ionic or covalent crystals. An example of an ionic crystal is salt (NaCl). Examples of covalent crystals are carbon, silicon, germanium, tin. They have four electrons in their outermost shell, and form tetrahedral (hcp) bonds.

11. Copyright © 2012 Pearson Education Inc. Crystal lattices and structures Example 42.4: It is an interesting exercise to calculate the potential energy of an ionic crystal. Consider such a crystal with alternating +e, –e charges with equal spacing a along a line. Show that the total interaction potential energy is negative, which means the crystal is stable. The quantity in parentheses is actually the expansion of the function ln(2), which is clearly a positive constant, so the overall potential energy is negative. In addition to ionic crystals and covalent crystals, a common type is metallic crystals. In this structure, one or more of the outermost electrons in each atom become detached from the parent atom, and are free to move through the crystal. The corresponding wave functions extend over many atoms. The freely moving electrons have many of the properties of a gas, and can be considered in the context of the electron-gas model (simplest of which is called the free-electron model).

12. Copyright © 2012 Pearson Education Inc. Types of crystals The figure below-left is indicative of the free-electron model—fixed ions and mobile electrons. Real crystals have defects, like the edge dislocation seen at right in two dimensions. Many materials are polycrystalline, having regions of uniform crystal structure in different orientations bonded together at grain boundaries.

13. Copyright © 2012 Pearson Education Inc. Energy bands The energy band concept, introduced in 1928, looks at how the outer energy levels of states in an atom vary with distance. When atoms are close enough together, the bands from one atom can join with another to permit lots of states in a closely spaced band of energy. Some bands are filled, some are empty, and in some materials some are partially filled. In an insulator, there is a large gap between occupied states and the “conduction” band where electrons can move in a metallic state. In a conductor, some electrons are in the conduction band, and therefore can conduct electricity.

14. Copyright © 2012 Pearson Education Inc. Free-electron model of metals The free-electron model assumes that electrons are completely free inside the metal, but that there are infinite potential-energy barriers at the surface. The 3D potential box from the previous chapter becomes the size of the entire lump of metal! For a cubic box, the energies are as we saw before: When L is large, these energies can be very small, and very close together. The density of states, dn/dE, is the number of states per unit energy range. Consider a sphere in “quantum-number-space” of radius. The number of states in the sphere is just the volume Because we are only dealing with positive numbers, though, we only have 1/8 th of a sphere. Also, each n-combination can have electrons with spin ±½, so twice as many. Finally, the number of states in the sphere-quadrant is The energy at the surface of the sphere is So writing L 3 = V, we have

15. Copyright © 2012 Pearson Education Inc. Fermi-Dirac distribution The Fermi-Dirac distribution f(E) is the probability that a state with energy E is occupied by an electron. The fraction of available states that are occupied at some temperature is denoted f(E). At absolute zero, all states up to some energy are filled (f(E) = 1), and that energy is called the Fermi energy E F0. As the temperature is increased, some electrons gain energy and go to higher states. The distribution of states is given by the Fermi-Dirac distribution: This function is graphed for increasing temperature as shown at the right. It is a bit tricky, because for some materials (like semiconductors) E F itself varies with temperature, but for solid conductors we can make the assumption E F = E F0.

16. Copyright © 2012 Pearson Education Inc. Electron concentration and free-electron energy We now have two quantities: the density of states between E and dE given by and the fraction of states filled as a function of temperature: We can combine these to find out the number dN of electrons with energies in the range dE as a function of temperature From our earlier expression at absolute zero n = N (the total number of electrons) and E = E F0 (the Fermi energy at absolute zero, from which we can solve for E F0 as

17. Copyright © 2012 Pearson Education Inc. Semiconductors A semiconductor has an electrical resistivity that is intermediate between those of good conductors and good insulators. Follow Example 42.9 using Figure 42.24 below.

18. Copyright © 2012 Pearson Education Inc. Holes A hole is a vacancy in a semiconductor. A hole in the valence band behaves like a positively charged particle. Figure 42.25 at the right shows the motions of electrons in the conduction band and holes in the valence band with an applied electric field.

19. Copyright © 2012 Pearson Education Inc. Impurities Doping is the deliberate addition of impurity elements. In an n-type semiconductor, the conductivity is due mostly to negative charge (electron) motion. In a p-type semiconductor, the conductivity is due mostly to positive charge (hole) motion. Follow the text analysis of impurities.

20. Copyright © 2012 Pearson Education Inc. n-type and p-type semiconductors Figure 42.26 (left) shows an n-type semiconductor, and Figure 42.27 (right) shows a p-type semiconductor.

21. Copyright © 2012 Pearson Education Inc. Photocell A photocell is a simple semiconductor device. See Figure 42.28 below.

22. Copyright © 2012 Pearson Education Inc. The p-n junction A p-n junction is the boundary in a semiconductor between a region containing p-type impurities and another region containing n-type impurities. Figure 42.29 below shows the behavior of a semiconductor p-n junction in a circuit.

23. Copyright © 2012 Pearson Education Inc. Currents through a p-n junction Follow the text analysis of currents through a p-n junction. Figure 42.30 below shows a p-n junction in equilibrium.

24. Copyright © 2012 Pearson Education Inc. Forward and reverse bias at a p-n junction Figure 42.31 (left) shows a p-n junction under forward- bias conditions. Figure 42.32 (right) shows a p-n junction under reverse- bias conditions.

25. Copyright © 2012 Pearson Education Inc. Transistors Follow the analysis of transistors in the text. Figure 42.33 (left) shows a p-n-p transistor in a circuit. Figure 42.34 (right) shows a common-emitter circuit.

26. Copyright © 2012 Pearson Education Inc. Integrated circuits Follow the text discussion of integrated circuits. Figure 42.35 (left) shows a field-effect transistor. Figure 42.36 (right) shows an actual integrated circuit chip.

27. Copyright © 2012 Pearson Education Inc. Superconductivity Follow the text summary of BCS theory and superconductivity.