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Unformatted text preview: Chapter 4. Phonons Lecture 8 1/25/11 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 Reminder: Homework #4 due Feb. 2 Tuesday • 5.1, 5.2, 5.4 • See end of lecture Quiz • Sketch transverse optic mode oscillation in diatomic linear lattice (at fixed time, t). • Sketch transverse acoustic mode oscillation in diatomic linear lattice. 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. Diatomic Linear Chain – contd. 2 1 2 2 2 [1 exp( )] [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 ) 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 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 Ka = 2 2 2 / w C M = 2 1 2 / w C M = optical acoustical 3 4 1 2 M M 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 u K v = Let’s note (u s – v s ) Why would we care?...
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This note was uploaded on 02/20/2012 for the course EE 123B taught by Professor Dianahuffaker during the Winter '11 term at UCLA.
 Winter '11
 DianaHuffaker

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