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Where and the force on each lattice atom harmonic

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Unformatted text preview: re 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 z-transform… 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 n-3 n-2 n-1 n n+1 un un+1 n+2 n+3 a un-3 un-2 un-1 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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