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 q and 1 e 1 S q 2 S q 2 n B 1 q r r r r where S q dx dt e iq x e i x x t 0 occupation factor T 1 and n B e 1 is the Bose b Show that q q and S q e S q 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 2 b M n S Q where b is the scattering of r r r the nucleus Q ki k f and r r i t S Q 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 q r 1 i qr R j q t aq e c c 2NM q 2 r where are the polarization vectors and labels the transverse and r r longitudinal modes Note that only Q u j appear in Eq 1 For simplicity r assume the modes are degenerate for each q so that we can always choose r r r one mode rwith parallel to Q Henceforth we will drop the label and r Qu j Then and treat Q u j as scalar products r r r r iQ R j R l F Q t e 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 cos jl isin jl 2NM q q where the Debye Waller factor 2W is given by 2W and n q 1 e Q2 1 2nq 1 2NM q q q r r r 1 jl 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 N V e 2W Q G G r r r Q2 n q 1 Q q G q r 2N M q q G r r r n q Q q G 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 of a given mode at energy q 6
View Full Document
Unlocking...