Ch. 22: Classical Theory of Harmonic Crystal

Transcription

1 C. : Clssl Toy o mo Cysl gl o ml moo o o os l s ld o ls o pl ollowg:. Eqlbm Pops p o ls d Islos Eqlbm sy d Cos Egs Tml Epso d lg. Tspo Pops T pd o lo Tm Fl o Wdm-Fz Lw pody Tml Cody o Islos Tsmsso o od. Io W do ly o Io Cysls Ils g o Lg g o X ys g o os T mo ppomo ssmpos:. Ios osll bo l ss m gd posos.. mplds o o moo m smll o spg. Fs Ts: Em osqs o l bo o os gy p-ws pols. Fo s l q Fo dym l P

2 Tylo pso mo ppomo s...! ] [ ] [ O z y q ] [ ] [ w Eqlly m δ o dym δ m Plggg [ ] [ ] [ ] s z y ] [ ] [ Ogl om od! s q dl o ll. Elos sly djs o slow g o poso db ppo.. Cosd s s mpl pms. m

3 dγ dγ l dγ L p gy dsy Cg o bls / / d d / / P ξ dp dξ dγ dγ p{ P / / T s d d d y d z q.. q ddξ p{ / } ξ / dγ d dp P q. m. / } q T υ T log & P dpd o T log d P Lw o C bs ol C J/mol K lmm. sm 5.7 υ T Copp.5 Gold 5.6 Ld 6. log & P: C 5.96 l/mol-k.9 J/mol-K l.9 Tgs.8 Z 5. y 8. I lssl p moo s ply mo low T so log d P lw sold b. ow lw oly wos ppomly g T. d qm oy o dsd low T bo! / m s υ s T s

4 oml ods o oom s L m K [ [ ] ] spg os b K s gbos m q o moo && K[ [ ] [ ] ] o-o Km K[ ] K os πj j: g K s : pos : -π/ o π/ oml mods smll K oms d d d d</. pg os K ll d G ll. K>G { C W ss m K G [ ] [ [ ] ] && K[ ] G[ [ ] ] && K[ ] G[ [ ] ] K G K G K G K G K G K ± s / K G } πj dm / K KG / K G wo bs: d - K G K m s / K G K G

5 o K G K ± s / K G / w K K / w K G << K K w K << G w smll K o op K G --> π/ K G ± K G KG K >> G G o s / op K o smll K G ± o K G m K G K K G KG os op K G --> π/ K >> G G o K op G o s / K op K G o s / op os / K K / / 5

6 6 oom s L m m && && / s ] [ m s l d. g os o y llowd. λ λ δ π π π log wlg lm ˆ ˆ j j δ os d Opl s oom ls os bs. Clls w bss ddo o os bs - opl bs. O o os bs s moos o o mss o bss d opl p bs s l moos mogs oms sm bss. -om bss moom l

7 Toy o Elsy L Com m moom l m { } { } m E τ E τ τ τ ll τ τ d E τ m τ E gd o τ E spd o 6 ls o : yy zz yz z y gd po oo s wol dpd oly o symm s X X m d X τ X τ τ τ [ τ τ 8 ll τ τ C. τ O τ τ m s symm oly dpd pms. ] Com L Els Toy ss s s o pss o p. j s o p d o - s w o j- do y z. ΔF Δlm Δ Fo obj o m soy s ompos j j. mb s smll ol g o dmso o ysl om qlbm. I w l s L -L/L s popool o l sss so p oss so by oo s lw. T Yog s modls E s ms o sss o sg o ml. E I gl sss d s ld og sss so modls C o qlly og ompl so. j jl l j C jl l l Els gy wo sod: C l ls j j j jcjl l j j l j l OTE: 99 ms jl l 7

8 Com Els Toy Oly 6 lms o d yy zz yz z 5 y 6 ms dpd C d sll old C d w smply d o 66. ow ls gy wold b wog. olo: dobl s s! Com Elsy yy zz yz z y yy zz yz z y / E yy / E zz / E yz z y / E / E / E / E / E / E / E / E yy zz yz z / E y 8

9 9 Com Toy o W Popgo d T & ρ W Lgg o ls mdm / ρ τ τ τ τ ρ d T & τ τ τ ρ & & L d δ τ solos log wlg sm s ds l. τ τ τ ρ T dpd o sp low mp oss w lo py. Cp. Qm Toy O mo Cysl Γ Γ d d Clsslly dp d d Γ Qm mlly E E E : soy ss g by mlo m P Fd s s o o. Loo o solo s: bd d Fo moom l

10 od Qzo smlssl m: pls qzd ld lssl ll qm m: ld qzd ommo los P d lo d o opos P [ ] δ δ [ ] [ ] P sl s O Qm Osllos. m gss o zd by poplo mbs o dpd osllos # poos E K E l d o osllos / l. m. q

11 g Tmp p T ls zo po gy g Tmp ldg m log-p << T T L oo m L T ompg w mo ob. Low Tmp p ods w >> T wll ob glgbly o sp. ow poos o os bs wll lwys ob sgly low T bs s. d / T T 8π. osd oly obos om os bs ysl s bss.. ppom dspso lo by log-wlg l om.. pl smmo w gl o ll -sp. Ll obo osd o s Z ywys. dpd o T T π p π 5 T p Wy s s o π? d π 5 g poo ps loy T dpd

12 Imd Tmp C EYE ITEPOLTIO CEE. pls ll bs w os bs w sm l dspso lo.. E b s gd o sp o ds s ol mb o mods qls ol mb o os ll os.. d 6π / T T π d Θ T Θ 9 π T 5 Θ Θ / T d T Θ Cold omp s w pm o g by mp. ow o ssm low T qo d ssm by mp o y w mp. Es odl Es ppomo by ppo. Es odl. Oly os mods d w by ppomo.. ssm os qy E o y opl bs ps. l. o op # opl bs op / E p E / T # lls T E / T

13 sy o oml ods dsy o poo lls d g δ 8π ol d g 8π lmm by modl π / < g > lbody do Epml d o dsbo o gy blbody do: s mp ss ol mo o gy ss. s mp ss p o dsbo o ss o so wlgs ollowg W s Lw λ m T m K so wlgs lssl oy pdd gy pm sowd o gy. Ts odo s lld lol sop ylg-js 8π T

14 lbody do Poo p T mb o sdg ws b md sd y s dl o mb o osllos ysl p o o o /. T gy o mod s qzd E T s o pp lm o w o poo s s s dl o os poo s mods low mp. y d π T π 5 -olzm T lw π d spl gy dsy Pl do lw