Lecture13

Lecture13

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Unformatted text preview: i (ni + 1 / 2 ) i ! We have a messy infinite sum of sums… ! Z = " " ... e #$ " !%i ( ni +1/2 ) i n1 =0 n2 =0 We can transform this into a slightly more friendly form & $ ! " !#i ( n+1/2 ) ) = , '% e * + i =1 ( n =0 We can transform this into a slightly more friendly form $ ' e! " !#i /2 = +% ( 1 ! exp(! " !#i ) ) i =1 & $ * 5 Energy of a harmonic solid (0 " -+% e! # !$i /2 < E >= ! ln / , & )2 "# . i=1 '1 ! exp(! # !$i ) * 1 "% = ! & ( ! # !$i / 2 ) ! ln (1 ! exp(! # !$i )) "# i=1 !!i = % !!i / 2 + 1 " e" # !!i i =1 $ (Assuming each oscillator has the same energy) 1 #1 & = 3N A !! % + !!/kT ( 2e "1 ' $ 6 Specific heat of a discrete harmonic oscillator solid: Einstein solid CV = ! "U "T e!!/kT " !! % CV = 3N A k $ ' kT & ( e!!/kT ( 1)2 # 2 V Nk Cv CV [ħω Ak] ­1 3N Ω 1.0 Dulong ­PeHt law recovered at high T 0.8 0.6 Low temperature limit sHll not reproduced Einstein CV decreases too fast as T approaches 0 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 T Ω ħω We haven’t considered the excitaHons as iden?cal...
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This document was uploaded on 02/04/2014.

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