Notes8 - ECE215A/Materials206A Fundamentals of Solids for...

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ECE215A/Materials206A Fundamentals of Solids for Electronics E.R. Brown/Winter 2008 1 NOTES 8: STATISTICAL MECHANICS OF LATTICE WAVES AND PHONONS Classical Average Energy Analysis of a Single Atom So far we have addressed the instantaneous mechanical behavior of lattice waves – classical and quantum mechanical. This led to an emphasis on the modal nature of every lattice wave, and the quanta of excitation for each mode – the phonon. Now we take the analysis one step further to understand the statistical mechanics, starting as before with the classical analysis and then developing the quantum description. We start with a monatomic linear lattice of atomic mass m, spacing a, and nearest neighbor (spring constant) C and ask the following question ? What is the energy of a single atom when a lattice wave is propagating. This will lead to a straightforward application of the Boltzmann formalism treating the atom itself as the subsystem. The total energy of the sth atom (chosen arbitrarily) is the kinetic plus half the potential with all neighbors (the half serving the requirement of counting each potential only once in summing to get the total energy) 2 11 2 22 dr s KE M M ss dt υ  →=   Accounting for nearest-neighbor interactions only, half the potential energy is () PE C r r C r r s s s −∆ = ∆ −∆ ++ And thus 2 2 1 s Ut M C r r dt =+ + (instantaneous) (1) To apply this we start with a unidirectional lattice wave of the (real) form: cos rA t s k a s ω ∆= . , so that sin s A ts k a dt ωω =− and ( ) 2 22 2 2 sin cos cos U t M A t ska CA t ska t ska ka k + The where the subscript k reminds us that the energy is just for the wave of k. The last term has the form 2 cos cos , with t ska ka αβ α α ω β  −− = − =  . So we can use the trigonometric identity cos cos cos sin sin βα −= + to write
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ECE215A/Materials206A Fundamentals of Solids for Electronics E.R. Brown/Winter 2008 2 () 2 22 2 2 11 sin cos cos 1 sin sin Ut k MA C A ω αα β α = +− + ( ) 22 2 2 2 2 sin cos cos 1 2cos cos 1 sin sin sin sin C A =− + + + To do the statistical mechanics, we do not need the instantaneous energy but rather the average over time. So we integrate over the period of wave, τ , and define 21 , o UU t kk f dt τ π ≡= = 2 sin ; cos ; 2 sin cos sin 2 0 oo o dt dt dt dt ττ τ ττ ααα == = = ∫∫∫ So, 12 1 2 2 2 cos 1 sin 2 cos 2cos 1 sin 44 2 1c o s 42 UM A C A k A C A k A C A k ωβ =+ +  + +   For a linear monatomic lattice, the lattice waves all have the acoustical form 2 2 o s C M . 111 442 A M AM A k ωω ω =+= And there are always two k values for the same ω , corresponding to the two different directions of wave flow. If we added two such waves of the same amplitude A, the time-averaged total energy would be U = M ω 2 A 2 ( 2 ) Statistical Mechanics We now assume the sth atom is in thermal equilibrium with the bath at temperature T . The Boltzmann pdf can be applied if the amplitude A
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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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Notes8 - ECE215A/Materials206A Fundamentals of Solids for...

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