Chapter 2 Reciprocal Lattice. An important concept for analyzing periodic structures

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1 Chpt Rcpocl Lttc A mpott cocpt o lyzg podc stuctus Rsos o toducg cpocl lttc Thoy o cystl dcto o x-ys, utos, d lctos. Wh th dcto mxmum? Wht s th tsty? Abstct study o uctos wth th podcty o Bvs lttc Fou tsomto. Rcpocl lttc s closly ltd to Fou tsomto Cosdto o how momtum cosvto wll b modd podc pottl Bsc kowldg: k k cos Wv vcto s pl wv ( k s( k k ˆ λ podc ucto

2 O dmsol cpocl lttc ( ( x x (x > > > ( ( s( cos( ( x x x x x x s c s c s c x s x c x O dmsol podc ucto Rd cocts Gx x x ( W hv us though ll tgs * Costt tht sus (x s l ucto G s o dmsol cpocl lttc Expd (x to Fou ss Th st o dsct pots (x s xpdd to ss o pl wvs, wth st o wv vctos G s. Ech pl wv psv th podcty o (x. Gx

3 l spc cpocl spc Rcpocl spc s Fou spc Rcpocl lttc pots o Bvs lttc dct th llowd tms th Fou ss o ucto wth th sm podcty s th Bvs lttc Rcpocl lttc dpds o th Bvs lttc, but dos ot dpd o th ptcul om o (x, s log s (x hs th podcty o th Bvs lttc But cpocl lttc dos ot dct th mgtud o th ch Fou tm, whch dpds o th xct om o (x. dx ( x Expd to th dmsol cs: x Th dmsol podc ucto ( Fou xpso: dx ( x Gx tht stss ( ( R G ( G G dv V G c cll ( G Volum o pmtv cll

4 Gl dto o cpocl lttc Bvs lttc R pl wv G G G v Th st o ll wv vctos tht yld pl wvs th podcty o gv Bvs lttc R s kow s ts cpocl lttc wth Alytclly, th dto s xpssd s Th st o G v G R stsyg G ( R G, o ll R Bvs lttc A cpocl lttc s dd wth c to ptcul Bvs lttc A st o vctos G v G R stsyg s clld cpocl lttc oly th st o s Bvs lttc Rcpocl lttc s Bvs lttc (pov th ollowg R 4

5 Costucto o cpocl lttc d Poo tht G s Bvs lttc b b ( ( A pmtv vctos o Bvs lttc Th cpocl lttc c b gtd by th pmtv vctos Apptly b j b δ j ( R b b b V ( volum o pmtv cll A bty vcto cpocl spc c b wtt s l combto o g g b gb gb g R To quly o cpocl lttc, Fo ll R g hs to b tgs Rcpocl lttc vcto G v b vb vb v s tg Apptly, t s lso Bvs lttc wth s ts pmtv vctos b 5

6 Th cpocl o th cpocl lttc Th cpocl lttc s Bvs lttc, o c costuct ts cpocl lttc, whch tus out to b othg but th ogl dct lttc. R Dct lttc G Rcpocl lttc G R o ll R O, vy R R G stss o ll Look o th cpocl o th cpocl lttc: K K G whch stss Th st o vctos R s subst o th st o vctos K R K G Fo y vcto tht dos ot blog to th dct lttc R: x x x At lst o x s o-tg Fo G b w hv G b x b x So dos ot blog to K So th st o vctos R s dtcl to th st o vctos K 6

7 Exmpls o cpocl lttc Smpl cubc Bvs lttc xˆ yˆ zˆ Rcpocl lttc b xˆ b yˆ b zˆ Also smpl cubc lttc Cubc ut cll wth dg o Fc ctd cubc ( ˆ y zˆ ( ˆ ˆ z x Pmtv vctos o Rcpocl lttc V ( 4 ( xˆ ˆ y b ( xˆ yˆ z ˆ b ( ˆ ˆ x y zˆ b ( xˆ yˆ ˆ z Thy th pmtv vctos o body ctd cubc lttc 4 Sz o th cubc cll: ( 7

8 Body ctd cubc Pmtv vctos ( ˆ ˆ x y zˆ ( ˆ ˆ x y zˆ ( ˆ ˆ ˆ x y z Pmtv vctos o cpocl lttc b ( yˆ z ˆ b ( ˆ x zˆ b ( xˆ y ˆ Thy th pmtv vctos o cc lttc Fcc d bcc cpocl to ch oth Bllou Zos Th Wg Stz pmtv cll o th cpocl lttc s kow s th st Bllou zo. 8

9 Fst Bllou zo o bcc stuctu Th st Bllou zo s closd by ppdcul bsctos btw th ctl lttc pot d ts st ghbos It s gul cd sold 9

10 Fst Bllou zo o cc stuctu Pmtv vctos o cc stuctu Bllou zos o cc stuctu Th cpocl lttc s bcc lttc

11 Lttc pls d cpocl lttc vctos Lttc pls: y pl tht cots t lst th ocoll Bvs ltc pots Du to tsltol symmty, y lttc pl wll cot tly my lttc pots, whch om two dmsol Bvs lttc Fmly o lttc pls: A st o plll, qully spcd lttc pls whch togth cots ll th pots o th th dmsol Bvs lttc. Thom: Fo y mly o lttc pls sptd by dstc d, th cpocl lttc vctos ppdcul to th pls, th shotst o whch hv lgth o. Covsly, o y d cpocl lttc vcto G, th s mly o lttc pls oml to G d sptd by dstc d, wh s th lgth o th shotst cpocl lttc vcto plll to G d Th thom s dct cosquc o ( Th dto o cpocl lttc G R ( th ct tht pl wv hs th sm vlu t ll pots lyg mly o pls tht ppdcul to ts wv vcto d sptd by tgl umb o wvlgth.

12 Poo o th st pt o th thom: Gv mly o lttc pls wth spto o d, lt ˆ b ut vcto oml to th pls. Lt s mk up wv vcto: K K ˆ d W d to pov tht s cpocl lttc vcto, d t s th shotst o tht dcto To quly o cpocl vcto, K R hs to b stsd o ll o th Bvs lttc W kow ths codto s stsd o th og o th Bvs lttc bcus K K Th must b tu o ll th lttc pots th lttc pl tht cots th og K Bcus s ppdcul to tht pl, d K R should b costt th pl K s podc ucto wth wvlgth λ d K th dcto o ts wv vcto λ d Th wvlgth hpps to b th spto o th pls Tho o ll th lttc pots th mly o pls, K R s stsd R So K s cpocl vcto, d w c wt t s G ˆ d

13 I th s oth vcto G shot th Th wvlgth o G s λ G > d G G < It s mpossbl o G to hv th sm vlu o two pls wth dstc shot th th wvlgth So G c ot b cpocl vcto Poo o th covs o th thom Gv cpocl lttc vcto, lt G b th shotst plll cpocl lttc vcto G Lt s costuct st o l spc pls (ot cssly lttc pls o whch Now w d to pov tht ths st o pls lttc pls d cots ll th Bvs lttc pots Ths pls must cot. Thy must b ll ppdcul to G d sptd by dstc d λ G Fo ll Bvs lttc vcto, G R must b tu o y cpocl vcto R So th st o pls must cot ll th Bvs lttc pots. G Lstly, w d to pov tht ch o ths pls cot lttc pots std o vy th o thm Suppos oly vy th o th pls cot lttc pots. Accodg to th st pt o th thom, th shotst G cpocl vcto ppdcul to th pls wll b,, whch cotdct wth ssmupto tht G s th d shotst cpocl vcto tht dcto

14 Rcpocl lttc vctos d Mll dcs o lttc pls Th mll dcs o lttc pl th coodts o th shotst cpocl lttc vcto oml to tht pl, wth spct to spcd st o pmtv cpocl lttc vctos. Thus pl wth Mll dcs h, k, l, s oml G hb to th cpocl lttc vcto Poo Cosd pl wth mll dcs h, k, l. kb lb G h b to tht pl. Lt s pov tht h h k k l l k b l b s th shotst cpocl vcto oml Fo y pot o pl, G A x x G x So Th t ollows tht G ( x A G ( x A G ( x A h : k : l x : x : x h k l A x A x A x Sc G s th shotst cpocl vcto ppdcul to th pl. Th should b o commo ctos btw h k l. Ths pmts stsy th ogl dto o mll dcs Tho h h k k l l 4