123B_1_EE 123B W11 lecture 6, chapter 4 part 2

123B_1_EE 123B W11 lecture 6, chapter 4 part 2 - Chapter 4....

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Unformatted text preview: Chapter 4. Phonons Lecture 6 1/20/11 Last Lecture: Chapter 4 Phonons Hookes Law force, displacement Newtons Law force, mass, acceleration Oscillation modes longitudinal, transverse Dispersion relation Group velocity Brillouin zone boundaries and limit Diatomic chains This Lecture Chapter 4 continued Reminder: Homework #3 due Feb. 2 Tuesday 4.1: For a monatomic chain, detemine total energy in an elastic wave. 4.3 For a diatomic basis, calculate wave amplitude ratio , u/v at K=/a . Extra what is the dispersion relation for a diatomic basis if m1=m2? Sketch. 4.5 For diatomic basis, find dispersion relationship at K=0 and K=/a. Phonon What are they? Propagation lattice vibrations that carry energy through a crystal Crystal can be considered an array of mass centers connected by springs Harmonic Oscillator Hookes law M oscillates with kinetic and potential energy. n=0 n=1 n=2 w k E M E n E n = ( + 1 2 29 mode number n = 2 = 2 , , = / , k = 2 We use Newtons law to derive (29 Energy in a Crystal Definition: Kinetic energy is the sum of the individual kinetic energies each of the form E 1 2 2 What is u? How do we relate F to E? How do we relate u to (amplitude to frequency)? Force between atoms s and s+1 is C(u s u s +1) Potential energy associated with the stretching of this bond is C(u s u s+1 ) 2 Newtons Law F Ma = 2 2 ( ) d r t M dt = v ( , ), r t = v X=0 X r x = v ( , ) ( , ) d F r t r t dx = v v ( ) is position of atom r t v In one dimension: potential energy ( 0) x = = ( ) x x k x = = arises from interaction between atom and rest of crystal. displacement 2 3 ( ) 1 ... r r r r = + + + + Liner nearest neighbor interaction Monatomic chain - Phonons Derive equation of motion S S+1 S-1 S+2 u S u S+1 u S-1 u S-u S- 1 u S- u S+1 2 1 1 2 ( 2 ) S S S S d u C u u u M dt +- +- = 2.) Equation of motion Hooke meets Newton 1 ( ) S S S S nn F C u u =- = , + 1 ( - + 1 29 + , -1 ( - -1 29 spring constant C S,S+1 =C S,S-1 =C 1.) Force- similar to Hookes law We assumed Nearest neighbors Uniform force constant u S ( t ) (- 29 d 2 u S dt 2 = - 2 3.) Write wave solution Find Dispersion Relation monatomic chain. 2 1 1 2 ( 2 ) S S S S d u C u u u M dt +- +- = d 2 u S dt 2 = - 2 C ( u S + 1 + -1- 2 29 = - 2 1 exp( )exp( ) S u u isKa iKa = 5.) Use wave solution: a: spacing between atomic planes K : the wavevector 4.) Combine equation of motion and wave solution: Acoustic, Optical Phonons - contd....
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123B_1_EE 123B W11 lecture 6, chapter 4 part 2 - Chapter 4....

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