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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 BoseEinstein 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 3D lattice with basis: For single atom basis in 3D, 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 kvector Creation and Annhilation Operators for Lattice Waves
Operators Operators for the Lattice Displacement
Operators We will use this for electronphonon scattering… Specific Heat with Continuum Model for Solid
Specific 3D continuum density of modes in dω : Specific Heat with Discrete Lattice
Specific
Density of Modes from Dispersion
1D continuum density of modes in dω : ω
ωm k ωm ω Specific Heat with Discrete Lattice
Specific
Density of Modes from Dispersion 3D 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 BoseEinstein 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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 Spring '03
 TerryOrlando

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