electron -ee mrw o center of atom CLASSICAL ELECTRON THEORY Lorentz' classical model for the dielectric function of insulators

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1 CLASSICAL ELECTRON THEORY Lorntz' claical odl for th dilctric function of inulator In thi odl th lctron ar aud to b bound to th nuclu ith forc obying Hook la. Th forc ar aud to b iotropic and daping can b includd through frictional forc proportional to th lctron vlocity. y cntr of ato v r o x lctron vg -E Schatic illutration of th lctron orbit around th cntr of th ato. Indicatd by thick arro ar th forc acting on th lctron: th cntriptal forc toard th cntr; th frictional forc in th dirction oppoit to that of th vlocity v; th xtrnal forc in th dirction oppoit to that of th xtrnal lctric fild E. Th quation of otion for an lctron i r + Gr + 0 r -E Thi i jut th Nton cond la, tating that a ti acclration qual th forc xrtd on th particl; hav collctd all tr xcpt th xtrnal forc on th lft hand id of th quation. Th firt tr i jut th a ti th acclration; th cond th frictional forc; th third th rtoring forc binding th lctron to th nuclu, placd at th origin. Th

2 Bo E. Srnliu 10: tr on th right hand id of th quation i th forc du to th xtrnal fild, E. Fourir tranforing thi diffrntial quation giv ( ) - ( ) r( ) - - ig + 0 E Thu r( ) E( ) ig ( ) Th inducd dipol ont pr lctron i p -r, that i, p( ) - ( - + ig) 0 1 E( ) Thu if au that thr i on lctron pr ato and n ato pr unit volu hav for th atoic polarizability a at ( ) - for th polarizability at a( ) na ( ) ( - + ig ), and for th dilctric function 0 1

3 Bo E. Srnliu 10: ( ) 1 + 4pna ( ) at p 1-4 n ig ( 0 ) pl 1 - ( ig) hr pl i th plaa frquncy. Thi tratnt can b gnralizd to inulator ith or than on lctron pr ato. W thn hav p ( )  j  j n j ( - j + igj ) pl, j ( - j + igj ) ;  j pl, j pl 4pn hr n j i th dnity of lctron ith ronanc frquncy j and n i th total dnity of lctron. Lt u tudy an xapl uing thi dilctric function ith th valu: Ïħ 0 4V Ô Ì( ħ pl ) 60( V ) Ô ħ 1 ÓÔ G V

4 Bo E. Srnliu 10: , ħ(v) Ral and iaginary part of a dilctric function in th Lorntz odl. Lt u no tudy th cattring of an lctroagntic av by a bound lctron. W au that th vlocity of th lctron ill not b rlativitic. Thn th agntic part of th Lorntz forc can b nglctd. Th lctric fild ill induc a dipol ont: p( ) - ( - + ig) 0 1 E( ) Th lctron ill b acclratd by th fild and ill radiat. Enrgy i aborbd fro th incidnt av by th lctron and i thn r-ittd into pac. Such a proc i calld th cattring of th lctroagntic av by th lctron. No, fro quation (9.15) and (9.16) hav for th por radiatd prunit-olid-angl by a ti dpndnt dipol

5 Bo E. Srnliu 10:5 [ dp p dw ] in q 4pc and for th total radiatd por [ p ] P c No uing th rul for Fourir tranfor hav ṗ( ) - + i ( 0 G) E( ) According to th ti-avrag product thor hav dp dw 1 4pc 1 È Í Í Î ig ( ) * È Í E Í ( - + i G Î 0 ) 0 in q È 1 Í E 8 c Í ( - ) + ( G) ÎÍ 0 0 in q p ( ) and P È 1 Í E c Í ( - ) + ( ) ÎÍ 0 0 G ( )

6 Bo E. Srnliu 10:6 Th cattring cro ction,, i rradiatd por incidnt por pr ara It i th quivalnt ara of th incidnt av front that dlivr th por rradiatd by th particl. Th dnoinator i th ti avrag of th Poynting flux S c E 0 8 p Thu hav for th diffrntial cattring cro ction d dw ( c ) - 0 in G ( ) + ( ) q and for th total cro ction 8p ( c ) - G ( ) + ( ) 0 Th xprion for th diffrntial co ction i valid for linarly polarizd light and q i th angl btn th dipol vctor and th dirction of th outgoing radiation. Th aborption lin ar up in th UV rang and in th viibl rang and blo hav that << 0. Thi an that

7 Bo E. Srnliu 10:7 Rayligh p Ê 8 ˆ Á Ë c 0 8p Ê 4p ˆ 1 Á Ë l 0 4 Rayligh cattring. Thi xplain hy th ky i blu and th unt rd. No inc th claical lctron radiu i r0 c on xpct claically to find th cattring cro ction to b 0 p 0 p Ê ˆ Ë Á r c for a fr, unbound, lctron. W find hr that ( ) ( - ) + ( G) 0 and that Rayligh 8 0 Ê ˆ Á Ë 0 4

8 Bo E. Srnliu 10: Full Rayligh Thoon / ħ (V) Th cattring cro ction ith th a paratr a bfor. W that for all nrgi th Rayligh cattring i valid. For high frqunci th cattring approach that for a fr, unbound, lctron. Evn highr up in nrgy, in th x-ray rang, hn th nrgy i of th ordr of th lctron rt nrgy thr ar quantu ffct odifying th rult. Th quantu chanical rult i knon a th Klin-Nihina forula. Ï8p Ê ˆ K.N. Ê Á - + << c ˆ Ë Ë Á Ô 1 ħ, ħ c Ì c Ô c È Ê ħ ˆ 1 p Á + Íln Î Ë, ħ >> c Ô Ó ħ c

9 Bo E. Srnliu 10:9 Drud' claical odl for th dilctric function of tal Th Drud odl for tal i obtaind fro th Lorntz odl by ltting th lctron b fr and not bound to th ato. Thi i obtaind by ltting 0 b zro: pl ( ) ig pl ( ig) ( ) - + Thi dilctric function ork rally ll. In th gnralizd Drud odl on lt G b frquncy dpndnt. In doing o on ha to lt G b coplx valud for to oby th Krar Kronig diprion rlation. It turn out that th ral part of G thn tay rathr contant all th ay up to th plaa nrgy and th iaginary part i of l iportanc. Thu th ipl Drud odl i valid all th ay up to th plaa nrgy. Lt u no tudy th cattring of an lctroagntic av by a fr lctron. W no hav p( ) - and 1 E( ) ( + ig) ṗ( ) E( ) ( + ig) If thr ar no frictional forc, no cattring againt ipuriti or ngligibl radiation daping, th xprion ar p( ) - E( )

10 Bo E. Srnliu 10:10 and ṗ( ) E( ) Thu hav for th diffrntial cattring cro ction d q r q dw Ê ˆ Ë Á in 0 in c and for th total cro ction 8p Ê Á Ë c ˆ 8p 8 r 0 0 Thoon p Ê 8 ˆ -4 Á ª c Ë c Thoon cattring.

11 Bo E. Srnliu 10:11 Dilctric function for a tal In a tallic yt thr ar alo bound charg contributing to th crning. Th Drud rult i thn odifid and hav: pl ( ) 1 - ( + ig) Æ pl ( ) ( ) - ; t tranport ti i 1 t [ + ( )] Th cond tr i rlatd to th dynaical conductivity ẽ( ) ( ) - 4pn p [ + ( t )] ( ) + 4 i ( ) i 1 hr ( ) n t 1 - it [ ] Th tatic conductivity i t ( 0) n hich you ar failiar ith. Dicuion: Hr hav nglctd th ontu dpndnc of th dilctric function. It i priibl in thi cour hr ar concrnd ith optical proprti or th intraction of lctroagntic av ith attr. Th ontu of th photon i uually vry all. W ar concrnd ith proc vry nar th frquncy axi of th q-plan.