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Unformatted text preview: Chapter 4: Crystal Lattice Dynamics Debye December 29, 2001 Contents 1 An Adiabatic Theory of Lattice Vibrations 2 1.1 The Equation of Motion . . . . . . . . . . . . . . . . . . 6 1.2 Example, a Linear Chain . . . . . . . . . . . . . . . . . . 8 1.3 The Constraints of Symmetry . . . . . . . . . . . . . . . 11 1.3.1 Symmetry of the Dispersion . . . . . . . . . . . . 12 1.3.2 Symmetry and the Need for Acoustic modes . . . 15 2 The Counting of Modes 18 2.1 Periodicity and the Quantization of States . . . . . . . . 19 2.2 Translational Invariance: First Brillouin Zone . . . . . . 19 2.3 Point Group Symmetry and Density of States . . . . . . 21 3 Normal Modes and Quantization 21 3.1 Quantization and Second Quantization . . . . . . . . . . 24 1 4 Theory of Neutron Scattering 26 4.1 Classical Theory of Neutron Scattering . . . . . . . . . . 27 4.2 Quantum Theory of Neutron Scattering . . . . . . . . . . 30 4.2.1 The DebyeWaller Factor . . . . . . . . . . . . . . 35 4.2.2 Zerophonon Elastic Scattering . . . . . . . . . . 36 4.2.3 OnePhonon Inelastic Scattering . . . . . . . . . . 37 2 A crystal lattice is special due to its long range order. As you ex plored in the homework, this yields a sharp diffraction pattern, espe cially in 3d. However, lattice vibrations are important. Among other things, they contribute to • the thermal conductivity of insulators is due to dispersive lattice vibrations, and it can be quite large (in fact, diamond has a ther mal conductivity which is about 6 times that of metallic copper). • in scattering they reduce of the spot intensities, and also allow for inelastic scattering where the energy of the scatterer (i.e. a neutron) changes due to the absorption or creation of a phonon in the target. • electronphonon interactions renormalize the properties of elec trons (make them heavier). • superconductivity (conventional) comes from multiple electron phonon scattering between timereversed electrons. 1 An Adiabatic Theory of Lattice Vibrations At first glance, a theory of lattice vibrations would appear impossibly daunting. We have N ≈ 10 23 ions interacting strongly (with energies of about ( e 2 / A )) with N electrons. However, there is a natural expansion parameter for this problem, which is the ratio of the electronic to the 3 ionic mass: m M ¿ 1 (1) which allows us to derive an accurate theory. Due to Newton’s third law, the forces on the ions and electrons are comparable F ∼ e 2 /a 2 , where a is the lattice constant. If we imagine that, at least for small excursions, the forces binding the electrons and the ions to the lattice may be modeled as harmonic oscillators, then F ∼ e 2 /a 2 ∼ mω 2 electron a ∼ Mω 2 ion a (2) This means that ω ion ω electron ∼ ˆ m M !...
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This note was uploaded on 01/10/2012 for the course PHYSICS 707 taught by Professor Electrodynamics during the Fall '11 term at LSU.
 Fall '11
 Electrodynamics

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