Ideal crystal : Regulary ordered point masses connected via harmonic springs

Transcription

1 Statistical thrmodyamics of crystals Mooatomic crystal Idal crystal : Rgulary ordrd poit masss coctd via harmoic sprigs Itratomic itractios Rprstd by th lattic forc-costat quivalt atom positios miima o PS vry atom movs aroud its quilibrium positio xampl: o-dimsioal crystal displacmt from quilibrium { ξ i } U N N N U 1 U x, x,..., x U(,,...,) x x x... 1 N i 1 x i 1 1 x i x

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3 U N N N U 1 U x, x,..., x U(,,...,) x x x... 1 N i 1 x i 1 1 x i x N N 1 U x, x,..., x U(,,...,) x x 1 N i i i 1 1 Harmoic approximatio U(ξ i ) is a quadratic fuctio rasoabl appraoximatio Forc costats i U(,,...,) dpds o th lattic paramtr fuctio of ρ = /N : U(, r) U x1, x,..., xn i pds o ρ 3N-6 idpdt vibratioal mods ~ 3N Coupld harmoic osc. 1 p m 1/ ad μ stads for ffctiv forc costat ad ffctiv rducd mass

4 Solvig th variatioal problm of atom cyrstal: trasformatio ito 3N idpdt harmoic oscilators. Frqucy of idividual oscilators dpds o masss, forc costats ad typ of th crystal (complicatd quatio) i N Frqucy of ormal mods dpds o dsity! Partitio fuctio of mooatomic crystal: 3N 6 U ( ; r ) Q, T q (o rotatioal ad traslatioal dgrs of frdom) vib, N 1 (atoms ar distiguishabl!)

5 ibratioal partitio fuctio harmoic oscilator 1 1 p m 1/ w 1 ibratioal lvl dgracy Zro rgy dfid as b b / b qvib ( T) 1 b/ b l Q d l q 1 1 T NT NQ 1 v v v v v / T T N, dt Q Q v hv / ibratioal tmpratur typically 1 3 K ust first trm ds to b cosidrd Populatio of vibratioal lvls: f T b( 1/ ) q vib Fractio of molcul i vibratioally xcitd stats: f T f T f b( 1/ ) b/ Qv /T qvib

6 Solvig th variatioal problm of atom cyrstal: trasformatio ito 3N idpdt harmoic oscilators. Frqucy of idividual oscilators dpds o masss, forc costats ad typ of th crystal (complicatd quatio) i N Frqucy of ormal mods dpds o dsity! Partitio fuctio of mooatomic crystal: 3N 6 U ( ; r)/ T Q, T q N 1 vib, (o rotatioal ad traslatioal dgrs of frdom) (atoms ar distiguishabl!) q 1 /T vib 1 /T 3N Q, T N 1 U ( ; r)/ T

7 Larg umbr of vibratioal mods (3N) cotiuous distributio from to ν max fi frqucy dsity g(ν)dν umbr of ormal vibratioal modls i a itrval (ν,ν+dν) T l Q T N, 1 /T 3N Q, T N 1 U ( ; r)/ T C T N, U( ; r) l l 1 ( ) Q g d T T Normalizatio coditio: g( ) d 3N W d a suitabl approximatio for g(ν); T proprtis ca b obtaid U( ; r) g( ) d 1 C g( ) d 1 Almost xact (harmoic approximatio oly) g(ν) is missig => arious approachs to fid g(ν)

8 I. Classical thrmodyamics ulog-ptit law ach vibratioal dgr of frdom cotributs basd o quipartitio thorm C 3N 3R 6 cal / dg. mol Wors for umrous crystals at high tmpraturs Fails at low tmpraturs Qualitativ failur at vry low tmpraturs (C approachs K as T 3 xprimtally) Silvr crystal

9 II. isti modl 197 Quatizatio of vibratioal rgy (similar to Plac modl of blac body) ach atoms vibrats aroud its quilibrium positio idpdtly of othr atoms 3N idpdt oscillators with th sam frqucy ν Usig g(ν): g( ) 3Nd (dlta fuctio) ν... Frvcy (isti s) 3N idpdt oscillators Spcifc for ach crystal dpds o th PS dtails U( ; r) g( ) d 1 C T C g( ) d 1 C 3N T 1 Q

10 isti tmpratur: Q Q / T Q C 3N T 1 Q / T Oly paramtr (isti tmpratur): Wors rmarably xcpt for vry low tmps. Q Q / : 3 T T C N T A. isti, A. Physi, (197) 18. Hat capacity of diamod Θ = 13 o K pdc of C o rducd tmpratur (Θ /T) is uivrsal for all crystals

11 III. by modl isti modl fails at low tmps Oscillator rgy dpds o frqucy T : Low rgy mods bcom importat Norma mod frqucy varis from do 1 13 Hz Blow ormal mods i 1- crystl (high ad low rgy modls dpictd blow) A mod havig th highst frqucy: wavlgth ~ a atoms mov agaist ach othr A mod with miimal frqucy atoms movs i th sam dirctio by: mods with wavlgth» lattic costat idpdt of matrial crystal bhavs as cotiuous lastic body Wav with amplitud A ad frqucy ω=πν ad movig i th dirctio : u( r, t) A r i( wt) wav vctor; π/λ v... locity of th wav u w / l Suprpositio of wavs movig i opposit dirctio: ir Stadig wav u A cos wt

12 To form a stadig wav - its imagiary part must b zro o th bordr (crystal dg): L x L y L z x y z p p p p L Frqucy dpds o u w / l p x y z L Numbr of wavs havig wavvctor smallr tha. F( ) p L L 6 p 6p 6p Numbr of vaws with wavumbr i itrval (, +d) w( ) d df d d d p istiguishig th dirctio Of th wav 1 p g( ) d 4 d 3 3 ut ul u u l p 4p g( ) d d 3 u ibratioal mods i th dirctio prpdicual (or paralll)

13 Itroducig avrag vlocity: 1p g( ) d d 3 u 3 1 u u u t l xact xprssio for low rgy mods by frqucy Maximal frqucy of th crystal follows from 3N 4p 1/3 u 9N g d d ( ) 3 g( ) d 3N C g( ) d 1 3 / 4 x T Q T x C 9N dx Q x 1 Q by tmpratur

14 by fuctio: 3 / 4 x T T Q T x Q 3 Q x 1 dx O-paramtr quatio, umrical solutio C T 3N Q For tmpratur approachig K: 4 1p T K : C N 5 T Q 3 A propr bhavior v for T gos to

15 Hat capacity as a fuctio of T/Θ sigl uivrsal curv

16 Alumiium Cadmium Chromium Coppr Gold Iro Lad Magas Nicl Platium 48 K 9 K 63 K K 165 K 47 K 15 K 41 K 45 K 4 K Silico 645 K Silvr 5 K Tatalum 4 K Ti (whit) K Titaium 4 K Tugst 4 K Zic 37 K Carbo 3 K Ic 19 K