chap4 - Chapter 4: Crystal Lattice Dynamics Debye December...

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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 Debye-Waller Factor . . . . . . . . . . . . . . 35 4.2.2 Zero-phonon Elastic Scattering . . . . . . . . . . 36 4.2.3 One-Phonon 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 3-d. 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. electron-phonon interactions renormalize the properties of elec- trons (make them heavier). superconductivity (conventional) comes from multiple electron- phonon scattering between time-reversed 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 Newtons 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 m 2 electron a M 2 ion a (2) This means that ion electron m M !...
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chap4 - Chapter 4: Crystal Lattice Dynamics Debye December...

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