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123B_1_EE 123B W11 lecture 8, chapter 4 part 3

123B_1_EE 123B W11 lecture 8, chapter 4 part 3 - Chapter 4...

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Chapter 4. Phonons Lecture 8 1/25/11
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Last Lecture: Chapter 4 – Phonons Phonons: Quantization Modes Momentum Scattering This Lecture – Chapter 4 continued Diatomic chains: Derived dispersion relation Acoustic and Optical Crystal directions
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Reminder: Homework #4 due Feb. 2 Tuesday 5.1, 5.2, 5.4 See end of lecture
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Quiz Sketch transverse optic mode oscillation in diatomic linear lattice (at fixed time, t). Sketch transverse acoustic mode oscillation in diatomic linear lattice.
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Diatomic Linear Chain 2 1 [1 exp( )] 2 w M u Cv iKa Cu - = + - - 2 2 [exp( ) 1] 2 w M v Cu iKa Cv - = + - Need to solve two equations with two variables. 2 1 1 2 ( 2 ) s s s s d u M C v v u dt - = + - 2 2 1 2 ( 2 ) s s s s d v M C u u v dt + = + - exp( )exp( ) s u u isKa iwt = - exp( )exp( ) s v v isKa iwt = - a: the distance between nearest identical planes Nearest neighbors in planes Equivalent C or C uv =C vu Equation of Motion: Want dispersion relationship.
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Diatomic Linear Chain – contd. 2 1 2 2 2 [1 exp( )] 0 [1 exp( )] 2 C M w C iKa C iKa C M w - - + - = - + - Solve in limit w(K): 1 Ka = 2 2 1 cos( ) 1 2 Ka K a 2245 - or 4 2 2 1 2 1 2 2 ( ) 2 (1 cos ) 0 M M w C M M w C Ka - + + - = or The two roots are: 2 1 2 1 1 2 ( ) w C M M 2245 + 1 2 2 2 2 1 2 C w K a M M 2245 + Optical branch Acoustic branch Resonates with e- and m- field Use quadratic equation: 1 2
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Diatomic Linear Chain – contd. Optical and acoustical branches of the dispersion relation for a diatomic linear lattice. 1 2 3 4 Frequency gap No modes oscillate between 1 2 2 2 C C w M M < < , K a π = cos 0 Ka = 2 2 2 / w C M = 2 1 2 / w C M = optical acoustical 3 4 1 2 M M
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Relative Displacement u/v Optical branch Get trend by consider K=0 (large λ) substitute 2 1 2 1 1 2 ( ) w C M M 2245 + 2 1 [1 exp( )] 2 w M u Cv iKa Cu - = + - - 2 2 [exp( ) 1] 2 w M v Cu iKa Cv - = + - into K v u v s v s u 1 s u + 1 s v + Atoms oscillate in opposite phase 2 1 M u v M = - 2 1 ( ); M u v M = - complex for 0 u K v = Let’s note (u s v s ) Why would we care?
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