Lecture13

Obey boseeinstein stahshcs total number of phonon

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Unformatted text preview: For this to be, β(E ­μ)>0 and so E ­μ>0. • Since lowest E=0, μ ≤ 0 9 Bose ­Einstein DistribuHon Not always true that parHcle number is conserved: Photons There is therefore NO condiHon to find the constant μ, so set it μ=0. Bose ­Einstein distribu?on for spin s par?cles n( E ) = 2s + 1 exp ( E / kT ) ! 1 N(E) 3 2 1 Photons are special…even though s=1 0 there are only 2 spin degrees of freedom. 0 1 2 E/kT 3 4 10 Phonons The elasHc vibraHons of a solid are quanHzed: phonons Phonons are spin = 0 excitaHons, and behave as bosons. Obey Bose ­Einstein staHsHcs. Total number of phonon modes is 3N (3 degrees of freedom) < U >= 3N A # n !!n e!!/kT " 1 “n ­space” Number of states between Emin and Emax=volume of shell 4πn2dn n 3D 4! n 2 dn = 3N 8" 0 3! < U >= 2 E 3 max $ fn ! 8 # 4" n 2 f dn n Emin nD is the maximum index (D stands for Debye) nD = (6 N / ! )1/3 !"n $ n exp(!" / kT ) # 1 dn 0 nD 2 11 Phonons For a confined parHcle in a box of linear d...
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This document was uploaded on 02/04/2014.

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