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Dr. Jie ZouPHY 13711 Chapter 43 Molecules and Solids

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Dr. Jie ZouPHY 13712 Outline Molecular bonds Bonding in solids Energy states and spectra of molecules Free electron theory of metals Band theory of solids Electrical conduction in metals, insulators, and semiconductors Semiconductor devices and superconductivity

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Dr. Jie ZouPHY 13713 Molecular bonds The bonding mechanisms in a molecule are fundamentally due to electric forces between atoms (or ions). Potential energy function that can be used to model a molecule: r: internuclear separation distance between the two atoms A and B are parameters that can be determined by experiments.

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Dr. Jie ZouPHY 13714 Plot of U(r) ~ r for a two-atom system Equilibrium separation: U(r) is a minimum and the two atoms are in stable equilibrium. At large separation distance: the slope is positive, corresponding to a net attractive force. At small separation distance: the slope is negative, corresponding to a net repulsive force. Binding energy: The additional energy the system has to be given to break up the diatomic molecule (so that r = ).

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Dr. Jie ZouPHY 13715 Classification of molecular bonding mechanisms Ionic bonds Example: Sodium Chloride (NaCl) Covalent bonds Example: H 2 molecule Van der Waals bonds Example: Condensation of inert gas atoms into the liquid phase Hydrogen bonds Example: DNA molecules

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Dr. Jie ZouPHY 13716 Example: Covalent bonding Ground-state wave functions 1 (r) and 2 (r) for two hydrogen atoms making a covalent bond ( ) (a) The atoms are far apart and their wave functions overlap minimally. (b) The atoms are close together, forming a composite wave function 1 (r) + 2 (r) for the system. The probability amplitude for an electron to be between the atoms is high.

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Dr. Jie ZouPHY 13717 Bonding in solids A Crystalline solid consists of a large number of atoms (ions) arranged in a regular array, forming a periodic structure. Classification of bonding in solids: Ionic solids. Example: Sodium Chloride (NaCl crystal) Covalent solids. Example: Diamond, silicon, germanium Metallic solids. Example: Copper, silver, sodium, etc.

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Dr. Jie ZouPHY 13718 Examples of bonding in solids NaClDiamond Metal

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Dr. Jie ZouPHY 13719 Energy states and spectra of molecules Total energy of a molecule: E = E el + E trans + E rot + E vib Rotational motion of molecules Vibrational motion of molecules Molecular spectra

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Dr. Jie ZouPHY 137110 Rotational motion of molecules E rot = (1/2) I 2. I = r 2, = m 1 m 2 /(m 1 + m 2 ), the reduced mass of the molecule Quantization of the magnitude of the molecule’s angular momentum J: rotational quantum number Allowed values of rotational energy: Energy separation between adjacent rotational levels:

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Dr. Jie ZouPHY 137111 Allowed Rotational transitions Selection rule: J = 1 For most molecules, transitions between adjacent rotational energy levels result in radiation that lies in the microwave range of frequencies (f ~ 10 11 Hz). Example 43.1: The J = 0 to J = 1 rotational transition of the CO molecule occurs at a frequency of 1.15 x 10 11 Hz. (a) Find the Moment of inertia of the molecule. (b) Find the bond length of the molecule.

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Dr. Jie ZouPHY 137112 Vibrational motion of molecules Frequency of vibration for the system: Allowed values of vibrational energy: Energy separation between successive vibrational levels:

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Dr. Jie ZouPHY 137113 Allowed vibrational transitions Selection rule: v = 1 Transitions between vibrational levels are caused by absorption of photons in the infrared region of the spectrum. Example 43.2: The frequency of the photon that causes the v = 0 to v = 1 transition in the CO molecule is 6.42 x 10 13 Hz. Find the force constant k for this molecule.

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Dr. Jie ZouPHY 137114 Molecular spectra Total energy of the molecule: Each energy level is indexed by the two quantum numbers v and J. Absorptive transitions between the v = 0 and v = 1 vibrational states: (1) J = +1 and (2) J = -1 Energies of the absorbed photons: (1) E = hf + (ħ 2 /I)(J+1), J= 0,1,2,… (2) E = hf - (ħ 2 /I)J, J=1,2,3…

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Dr. Jie ZouPHY 137115 Absorption spectrum of the HCl molecule Quick Quiz: There is a gap between the two sets of peaks. Why?

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Dr. Jie ZouPHY 137116 Homework Chapter 43, P. 1434, Problems: #3, 8, 9, 14, 17.

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