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and the force on each lattice atom Harmonic Matrix
Harmonic
Spring Constants Between Lattice Atoms Harmonic Matrix: Dynamics of Lattice Atoms
Dynamics Force on the jth atom (away from equilibrium)… Solutions of Equations of Motion
Solutions
Convert to Difference Equation Time harmonic solutions… Plugging in, converts equation of motion into coupled difference equations: Solutions of Equations of Motion
Solutions We can guess solution of the form: This is equivalent to taking the ztransform… Solutions of Equations of Motion
Solutions
Consider Undamped Lattice Vibrations
Consider
Lattice We are going to consider the undamped vibrations of the lattice: Solutions of Equations of Motion
Solutions
Dynamical Matrix Solutions of Equations of Motion
Solutions
Dynamical Matrix Strain in a Discrete Lattice
Strain
Example of Nearest Neighbor Interactions
equilibrium
n3 n2 n1 n n+1 un un+1 n+2 n+3 a un3 un2 un1 un+2 un+3
strained Strain in a Discrete Lattice
Strain
Example of Nearest Neighbor Interactions Strain in a Discrete Lattice
Strain
Example of Nearest Neighbor Interactions
Harmonic matrix: Dynamical matrix: Strain in a Discrete Lattice
Strain
Example of Nearest Neighbor Interactions ω k=  π/a k=π/a k Strain in a Discrete Lattice
Strain
Example of Nearest Neighbor Interactions 2nd Brillouin zone k=2π/a 1st Brillouin zone k=π/a A ω 2nd Brillouin zone k=π/a B k=2π/a k From what we know about Brillouin zon...
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This document was uploaded on 03/19/2014 for the course EECS 6.730 at MIT.
 Spring '03
 TerryOrlando

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