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MIT OpenCourseWare http://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 1.) (a) Due: February 12, 2004 Prove the finite temperature version of the fluctuation dissipation theorem χ ′′(q,ω ) = and 1 − βω (e −1)S(q,ω ) , 2 S (q,ω ) = −2( n B (ω ) + 1) χ ′′(q,ω ) , € r r r r where S (q,ω ) = ∫ dx dt e−iq −x e iωx ρ( x,t ) ρ (0, occupation € factor. )T −1 and n B (ω ) = (e βω −1) is the Bose € € (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 r r r 2 momentum ki to k f is given by (2πb / M n ) S Q,ω where b is the scattering of r r r the nucleus Q = ki − k f and ( ) € r r iωt € S Q,ω = ∫ dt e F Q,t ( ) € ( ) € where r r r r r −iQ⋅ r (t ) F Q,t = ∑ e j e iQ⋅ rl (0) € ( ) j,e (1) T r r r r r and rj (t) is the instantaneous nucleus position. Write rj = R j + u j , where R j r are the lattice sites, € and expand u j in terms of phonon modes r r u j = ∑∑ λα € α € q r 1 i( qr ⋅ R j −ω q t ) aq e +€ c.c. 2NMω q € € (2) r where λα are the polarization vectors and α labels the transverse and r r longitudinal modes. Note that only Q ⋅ u j appear in Eq. (1). For simplicity, r assume the α modes are degenerate for each q so that we can always choose r r r € one mode rwith λα parallel to Q . Henceforth we will drop the α label and λα r Qu j . Then and treat Q ⋅ u j as scalar products € € r r r r −iQ⋅ ( R j −R l ) €e € F Q,t = ∑ € F jl (t) ( ) jl € where € F jl (t) = e € −iQu j (t ) iQul (0) e T . a) Show that € F jl (t) = e −iQ ( u j (t )−ul (0)) e 2[ 1 Qu j (t ),Qul (0)] (3) T Furthermore, for harmonic oscillators, the first factor can be written as € e −iQ ( u j (t )−ul (0)) T =e − 12 Q 2 ( u j (t )−ul (0)) 2 (4) T b) Using Eqs. (1–4) show that € F jl (t) = e −2W Q2 1 exp ∑ ( 2nq + 1) cosθ jl + isinθ jl 2NM q ω q ( where the Debye-Waller factor 2W is given by € 2W = ( and n q = 1 e Q2 1 ∑ (2nq + 1) 2NM q ω q βω q € € € r r r −1 , θ jl = −ω q t + q ⋅ R j − Rl . ) ( ) ) (5) c) Expand the exp factor in Eq. (5) to lowest order and show that (V* is the volume of reciprocal lattice unit cell) r r S (Q,ω ) = N V * e−2W ∑δ Q − G G r r r Q2 + ∑ ( n q + 1)∑δ Q − q − G δ (ω − ω q ) r 2N M ω q q G r r r + n q ∑ δ Q + q − G δ (ω + ω q ) G ( ) ( ( ) ) d) Discuss the interpretation of various terms in Eq. (6) . € e) Even though we did not compute it explicitly, what experiment would you propose to measure the polarization vector λα of a given mode at energy ω q ? € € (6)