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Chapter 29 MOLECULAR AND SOLID-STATE PHYSICS GOALS When you have mastered the content of this chapter, you will be able to achieve the following goals: Definitions Define each of the following terms, and use each term in an operational definition: ionic bonding extrinsic semiconductor covalent bonding semiconductor fluorescence superconductivity phosphorescence nuclear magnetic resonance bioluminescence Band Theory of Solids State the band theory of solids, and use it to explain the optical and electrical properties of different solids. Molecular Absorption Spectra Explain the basis of vibrational and rotational absorption spectra of molecules. Solid-State Problems Solve problems involving vibrational and rotational energy states of molecules and the band theory of solids.

PREREQUISITES Before beginning this chapter you should have achieved the goals of Chapter 21, Electrical Properties of Matter, Chapter 27, Quantum and Relativistic Physics, and Chapter 28, Atomic Physics.

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Chapter 29 MOLECULAR AND SOLID-STATE PHYSICS 29.1 Introduction What factors determine the structure of molecules? Molecular construction is important in understanding the chemical properties and reactions of matter. The field of organic chemistry involves the study of specific atoms and the way they are joined together to form a molecule. The structure of the molecules is closely related to the chemical behavior of materials. In this chapter we will study the physical basis for molecular bonding and emission and absorption spectra of molecules. Have you ever wondered why some solids are opaque and mirrorlike while others are translucent or transparent? What is the physical basis for the optical properties of solids? Have you noticed the difference in the thermal conductivity between stainless steel tableware and silver tableware? Have you noticed that good electrical conductors are usually good thermal conductors? In this chapter we will develop a model that will enable us to understand the behavior of solids in a qualitative way.

29.2 Molecules and Bonding Molecules are stable configurations of atoms. As in other naturally stable systems, the stability of molecules indicates that the energy of the molecule is lower than the energy of the system of separate atoms. We can formulate the following guide: if interacting atoms can combine and thereby reduce the total energy of the system, then the atoms will form a molecule. What can you conclude about monatomic gases (such as helium, He) as compared with diatomic gases (such as hydrogen, H2)? A useful model for describing diatomic molecules is the "spring model," in which two atoms are bound by a Hooke's law force. The electronic states of atoms determine the nature of the bonding that occurs in the formation of molecules. As atoms are brought together there are three limiting situations that can occur. These define the pure cases of chemical bonding Figure 29.1.

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1. NO BOND IS FORMED BETWEEN INTERACTING ATOMS If all of the lowest electron states of the interacting atoms are filled, the atoms repel each other when these electronic shells overlap. This is a quantum force resulting from the Pauli exclusion principle, which does not allow two electrons to occupy the same quantum state. Helium atoms behave in this way. 2. IONIC BOND IS FORMED In this case one or more electrons are stripped from one atom and captured by another atom. The first atom is left with a positive charge and the second atom becomes negatively charged. They are held together by the electrostatic attraction of these two ions. Sodium chloride, Na+Cl-, is an example of an ionic compound. The sodium atom gives up an electron to the chlorine atom, and the resulting Na+ and Cl- attract each other to form the NaCl system. Ionic compounds are usually soluble in water. Can you give a plausible physical explanation for this solubility, remembering that the dielectric constant of water is about 80? 3. COVALENT BOND IS FORMED In this case the atoms share electrons. This sharing produces a high probability electron distribution between the atoms. These shared electrons produce an attracting force between the atoms. Hydrogen, H2, is an example of nonpolar covalent bonding, and HCl is a polar covalent bond. A nonpolar bond (such as that in H2, Cl2) is electrically symmetrical producing no external electric field. In general, nonpolar molecules are usually gases or liquids at room temperature. If nonpolar, covalent molecules form solids, they are soft and easily vaporized. What physical explanation can you give for this observation? On the other hand, polar molecules exhibit relatively strong external electric fields due to their nonsymmetrical charge separation. For example, in HCl the shared electrons are closer to the chlorine nucleus than to the hydrogen nucleus, thus making a polar molecule as shown in Figure 29.1. Polar substances usually have higher melting and boiling points than nonpolar substances. Hydrogen gas, H2, represents a typical molecule with a covalent bond, and sodium chloride, NaCl, represents a typical molecule with an ionic bond, there are many molecules that represent intermediate bonding, with some characteristics of these two pure cases.

29.3 Absorption Spectra for Molecules The way in which a substance absorbs light provides information about the molecular structure of the substance. The spectrophotometer is an instrument designed to measure and record the absorption spectra of molecular and atomic samples. The details of the spectrophotometer were discussed in Chapter 28. As with atoms, the absorption spectra of molecules are the result of photon absorption by the system (in this case the molecules of the sample). The absorption spectra of molecules are more complex than those of atoms because in addition to the electronic energy transitions available, there are additional vibrational and rotational energy transitions available in molecules. The quantum equation that applies to the photon absorption process is given by Planck's equation, E = hƒ

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where h is Planck's constant = 4.13 x 10- 15 eV-sec and f is the frequency of the photons absorbed. The output of the spectrophotometer is a plot of the percent of light transmitted as a function of wavelength. The absorption of visible and ultraviolet wavelengths is the result of electronic transitions from a lower to a higher energy level in the molecules of the sample. These energy transitions are the order of electron volts in magnitude. The vibrational and rotational energy states of the molecules are quantified just as the electronic states are. These energy levels are much closer together, and the energy differences are the order of 10-2 eV for the vibrational states and 10-4 eV for the rotational states. The wavelengths absorbed by the transitions between vibrational states lie in the infrared region 1.5 to 30 microns (1m = 10-6 m), while the absorption that arises from transitions between rotational states occurs in the infrared and microwave regions (wavelengths from 30 m to 1 cm).

29.4 Energy Levels for Vibrational States The energy level differences for vibrational states depends upon the stiffness of the bond (analogous to the spring constant of a spring) and upon the effective mass of the vibrating atoms making up the molecule. For a diatomic molecule we can use a dumbbell model, Figure 29.2. The equation for the energy levels for the vibrating diatomic dumbbell is: Evib = (n + 1/2)hf = (n + 1/2) h/2π)(k/m)1/2

(29.2)

where k is the effective spring constant of the bond, m is the effective mass of the molecule given by m = M1M2/(M1 + M2) where M1, and M2 are the atomic masses of the two vibrating atoms in kilogram and n is the vibrational quantum number (n = 0, 1, 2, ...). The expression for f in an SHM system was developed in Chapter 15.

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  EXAMPLE

Find E0 (the ground state) for the vibrational level of carbon monoxide, CO. For CO molecules, k = 1870 N/m and m = (12 x 16 / (12 + 16)) x 1.67 x 10-27 kg m = 1.14 x 10-26 kg Thus, ƒ = 1 / (2π)(k/m)1/2 = 1/(2π (1870 N/m / 1.14 x 10-26 kg)1/2) ƒ = 6.45 x 1013 Hz E0 = (½) hƒ = 3.3 10-34 J-sec x 6.45 x 1013 Hz = 21.29 x 10-21J = 0.13 eV

29.5 Energy Levels for Rotational States The energy levels for rotational states depend upon the effective mass of the molecules and the separation of the atoms making up the molecule. The equation for the rotational energy level is: Erot = J(J + 1)(h/2π)2/2mR2) = J(J + 1)(h/2π)2/2I

(29.3)

where m is the effective mass, R, the separation of atoms, and J, the rotational quantum number (J = 0, 1, 2, ...). Note that mR2 is the moment of inertia, I, of the molecule about an axis through the center of mass. EXAMPLE Find the energy difference (J = 0 to J = 1) for rotational transitions in CO molecules. From the previous problem we have m = 1.14 x 10 26 kg. For CO, R = 1.13 x 10-10 m. For J = 1, E1 = 1(1 + 1)(h/2π)2/2mR2 = (1.05 x 10-34 J-sec)2/(1.14 x 10-26 kg x (1.13 x 10-10 m)2) E1 = 7.57 x 10-23 J = 4.75 x 10-4 eV E0

=0

∆E = 4.73 x 10-4 eV

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29.6 Identifying Molecular Structure In practice absorption spectra give the wavelengths of the absorbed photons. This information is then used to calculate the energy differences for the vibrational and/or rotational states. The energy differences can then be used to calculate the effective spring constants for the bonds and the separation of the atoms making up the molecules. (The masses of the constituent atoms are known quantities in this case.) In complex molecules there are different bonds and different separations in different directions. These considerations lead to a complicated set of energy levels that can be used with the spectrophotometer data to analyze the molecule's structure. No two molecules have the same absorption spectra. The vibrational and rotational spectra are unique characteristics of the molecules. Certain functional groups are found to have characteristic absorption band spectra that can be used to identify new molecular structures and to help distinguish between isomers (compounds involving the same kinds of atoms, but having different structure.)

29.7 Fluorescence and Phosphorescence The phenomena of fluorescence and phosphorescence arise from the complexity of molecular energy level systems and interactions. In each case the system is excited by external radiation-usually visible or ultraviolet light. In the case of fluorescence, the system gives up some of its energy (through vibrational excitation) in radiationless collisions with other molecules. The molecule then emits a photon from a lower vibrational state, and thus a longer wavelength photon is emitted. In phosphorescence there is a radiationless relaxation from one excited state to another excited state with a long lifetime (usually called a metastable state). The subsequent decay to the ground state is relatively slow and gives a characteristic afterglow associated with phosphorescence.

29.8 Solids The solid state of matter is characterized by its relative rigidity and fixed volume. We can use as a model for a solid a system of atoms, molecules, or ions rigidly held close together. The binding forces in solids may be ionic, covalent, van der Waals, or metallic. Ionic binding results in solids where positive and negative ions are involved as in ionic molecules, and the binding energy is electrostatic energy. An example of an ionic solid is NaCl (table salt). The solid state ionic bond is strong, and ionic solids are usually hard crystals with high melting points. Would you expect ionic solids to be brittle or ductile? Ionic crystals are usually soluble in polar liquids. Covalent solids result from the sharing of electrons just as in the molecular covalent bond. Examples of covalent crystals are the semi-conductor silicon and germanium. The solid-state covalent binding force is very strong, and these solids are very hard (diamond, for example, the hardest substance known, is a covalent solid) and have high melting points. Covalent solids are insoluble in most liquids.

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The van der Waals force in solids results when the atoms involved have dipole moments (nonsymmetric charge distribution) and display primarily dipole-dipole interactions Figure 29.3. This dipole-dipole interaction results in a weak binding force. Molecular crystals having van der Waals bonding have low melting points and boiling points. Solid methane (melting point = -259°C) is an example of such a solid.

The metallic bond results from the attraction of the free electrons (made up of the valence electrons of the atoms) that are shared by all of the positive ion cores. This gives rise to a very strong metallic bond. This metallic bond is similar to the covalent bond but it is unsaturated-that is, it could accommodate more electrons with little increase in the energy of the bond. The unsaturated nature of the metallic bond makes it possible to form mixtures of metals called alloys that vary in properties as the composition is changed. Modern technology has made good use of alloys with special electrical, magnetic, and thermal properties. Can you think of alloys that are used for specific applications?

29.9 Electron Behavior in Solids Classical physics explained the electrical conductivity of solids through the use of the electron gas model. According to this classical model good conductors have loosely bound electrons that move through the solid. The resistance of the conductor is accounted for by the collisions between the electrons and the vibrating positive ion cores. Using this model, we can explain why the resistance of a conductor should increase as the temperature is increased. In this classical model the insulators are represented as systems with tightly bound electrons. This model provides a qualitative picture of electrical phenomena in solids, but it fails to give good quantitative predictions. Quantum physics provides us with a much more comprehensive understanding of the electron behavior in solids. The question that needs answering is what happens to the quantized energy levels of the atoms as they condense into the solid state? The exact solution of this N-atom (where N is in the order of 1023) problem is not known exactly, but quantum physics can be used to give us a model for solids. As the N atoms condense into the solid state, the individual electronic states form bands of closely spaced energy levels. The energy levels are so close within a band that for practical purposes they can be treated as a continuum. But just as with the atomic energy levels, there are certain forbidden energy gaps between allowed energy bands. According to

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the Pauli exclusion principle there is a set number of electrons in any band. The band theory model applied to a conductor, semiconductor, and an insulator is illustrated in Figure 29.4. The highest energy, completely filled band is called the valence band. The next higher energy band, which is at most partially filled, is called the conduction band.

29.10 Conductors For a conductor, such as silver, the band of energy levels that contain the conduction electrons, called the conduction band, is only partially filled. This means that there are empty energy states very near (