MIT OpenCourseWarehttp://ocw.mit.edu 8.512 Theory of Solids II Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.€€€€€€€€€€€€€€€€€8.512 Theory of Solids Problem Set #1 Due: February 12, 2004 1.) (a) Prove the finite temperature version of the fluctuation dissipation theorem χ′′(q,ω)= 1 e−βω −1)S q,ω) ,((2 and S q( ,ω)= −2(nB (ω) + 1)χ′′(q,ω) , r where S q,ω)= dx r dt e−iq r −x r eiωx ρ(x,t)ρ(0,) and nB ( )ω=(eβω −1)−1 is the Bose( ∫ Toccupation factor. (b) Show that χ′′ q,ω)= −χ′′ −q,−ω) and S −q,−ω)= e−βω S q,ω) . In terms of the scattering probability, show that this is consistent with detailed balance. ( (( ( 2.) Neutron scattering by crystals. We showed in class that the probability of neutron scattering with momentum kr i to kr f is given by (2πb / Mn )2 S(Qr ,ω) where b is the scattering of rrr the nucleus Q = ki − kf and S(Qr ,ω)= ∫ dt eiωtF(Qr ,t) where rrrrr rj (t )rl (0) j,eT −iQ⋅F ( )= iQ⋅∑ (1) Q,t e eand rr j (t) is the instantaneous nucleus position. Write rr j = R r j + u r j , where R r j are the lattice sites, and expand u r j in terms of phonon modes u r j = ∑∑λr α 1 aqei(q r ⋅R r j −ωqt)+ c.c. (2)2NMω α q q€€€€€€€€€€€€€€€€r where λα are the polarization vectors and α labels the transverse and longitudinal modes. Note that only Qr ⋅ u r j appear in Eq. (1). For simplicity, assume the α modes are degenerate for each q r so that we can always choose rr r one mode with λα parallel to Q. Henceforth we will drop the α label and λα and treat Qr ⋅ u r j as scalar products Qu j . Then r rrr( ) Fjl (t)−iQ⋅ Rj −RlF ( )= ∑ Q,t e jl where −iQu j (t )iQul (0) .Fjl (t) = e eT a) Show that −iQ u j (t )− ul (0) ) e 1 [Qu j (t ),Qul (0) ] (3) ( 2Fjl (t) = e T Furthermore, for harmonic oscillators, the first factor can be written as − 1 Q 2 (uj (t )− ul (0) )2 2−iQ u j (t )− ul (0) ( ) T (4) e = e T b) Using Eqs. (1–4) show that ) Q2 1 (( −2W + 1)cosθ jl + isinθ jl ∑ (5) (t) = e 2nFjl exp q2NM ω qq where the Debye-Waller factor 2W is given by 2W = Q2 ∑ 1 (2n + 1)2NM ωq qq and n = 1 (e βωq −1) , θ jl = −ωqt + q r ⋅ (R r j − Rr l ) .q€€€c) Expand the exp factor in Eq. (5) to lowest order and show that (V* is the volume of reciprocal lattice unit cell) −2W ∑δ(rr) S Q,ω)= N V * e( Q − G G Q2 (nq + 1)∑δ(M ωr r G) r δ ω −ω( ) ∑ (6) Q − q − + q2N rGqq q ) Q + rrr( )δ ω + ω(G δ q −+ n∑q G d) Discuss the interpretation of various terms in Eq. (6) . e) Even though we did not compute it explicitly, what experiment wouldyou propose to measure the polarization vector λα of a given mode at energy ωq
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