lecture_10_11c notes

Approach quantize the amplitude of vibration for each

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Unformatted text preview: each mode ? Approach: • Quantize the amplitude of vibration for each mode • Treat each quanta of vibrational excitation as a bosonic particle, the phonon • Use Bose-Einstein statistics to determine the number of phonons in each mode Simple Harmonic Oscillator Simple ψ (x) E 2 U ( x ) = 1 mω 2 x 2 2 E3 = 7 h ω 2 n=3 E2 = 5 hω 2 n=2 n =1 n=0 E1 = 3 h ω 2 E0 = 1 hω 2 x Hamiltonian for Discrete Lattice Hamiltonian Potential energy of bonds in 3-D lattice with basis: For single atom basis in 3-D, µ & ν denote x,y, or z direction: Hamiltonian for Discrete Lattice Hamiltonian Plane Wave Expansion The lattice wave can be represented as a superposition of plane waves (eigenmodes) with a known dispersion relation (eigenvalues)…. σ denotes polarization Commutation Relation for Plane Wave Displacement Commutation …commute unless we have same polarization and k-vector Creation and Annhilation Operators for Lattice Waves Operators Operators for the Lattice Displacement Operators We will use this for electron-phonon scattering… Specific Heat with Continuum Model for Solid Specific 3-D continuum density of modes in dω : Specific Heat with Discrete Lattice Specific Density of Modes from Dispersion 1-D continuum density of modes in dω : ω ωm k ωm ω Specific Heat with Discrete Lattice Specific Density of Modes from Dispersion 3-D continuum density of modes in dω : Cu Specific Heat of Solid Specific How much energy is in each mode ? Approach: • Quantize the amplitude of vibration for each mode • Treat each quanta of vibrational excitation as a bosonic particle, the phonon • Use Bose-Einstein statistics to determine the number of phonons in each mode Specific Heat of Solid Specific How much energy is in each mode ? And we are done…...
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