HW4_solns_2

HW4_solns_2 - ECE215A Winter Quarter 2008 Solutions to HW...

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ECE215A Winter Quarter 200 8 Solutions to HW #4 , part 2 5 4). Acoustical phonons in a two-dim lattice (a) Excluding zero point term, and assuming the two acoustic waves have the same velocity, () ( ) 00 22 1 DD k kk kT Dd Un D e ωω ω =< > = ∫∫ = = = 2 polarizations, one longitudinal, one transverse For each polarization ( ) 2 2 2 2 2 L dN k dk d d A D dkd dk d dk π ωπ ωυ υ  == =  where we have k velocity of sound =→ and D is defined by 2 24 D C A A ND d d υπ === . Thus, 2 2 0 1 D kT Ad U e πυ = = = . Define 3 2 0 , 1 D x D D x AkT xdx xk T U x ek T =⇒ = = = = . (b) In the limit of low temperature, T →∞ = . And using the clue 2 0 2.40, 1 x e we find ( ) ( ) 33 3 2.40 2.40 4 9.6 BC B C B N kT N kT U ≅= = = 32 7.20 B v dU Ak T C dT ⋅⋅ ≈= = (c) In the limit high temperature 11 1 x ex So, 2 2 2 0 D x BB B D D x Ux d x = . But from definition of D in two dim in (a), we have 2 2 4 2 C DC B N UN k T A = and 2 vC B CN k dT (no surprise; this is just 2/3 of Dulong-Petit law in 3 dim). 5). Phonon propagation in quasi-two-dimensional crystal (a) Phonon energy is found by assuming only phonons in the quasi-two-dim plane are important.
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This note was uploaded on 01/17/2012 for the course ECE 215A taught by Professor Brown during the Winter '08 term at UCSB.

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HW4_solns_2 - ECE215A Winter Quarter 2008 Solutions to HW...

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