ECEN 5005 Crystals, Nanocrystals and Device Applications Class 14 Group Theory For Crystals

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1 ECEN 5005 Cryta Naocryta ad Dvic Appicatio Ca 14 Group Thory For Cryta Spi Aguar Motu Quatu Stat of Hydrog-ik Ato Sig Ectro Cryta Fid Thory Fu Rotatio Group 1

2 Spi Aguar Motu Spi itriic aguar otu of ctro (or othr icrocopic partic) which i aociatd with it agtic ot. - Thi do NOT a ctro ar actuay piig. Spi tr ito th Haitoia through (1) pi-orbit itractio tr ζ( r ) () Za itractio tr ( / ) B agtic fid B. wh thr i a xtra Th pi aguar otu oprator bhav iiary to th orbita aguar otu oprator ad ca b charactrizd by quatu ubr ad. χ ( + 1) z χ χ whr χ i th pi wavfuctio. Th oy aowd vau for i ½ ad ca b ithr +½ or ½. Th tat with +½ i oti cad th pi-up tat ad ½ th pi-dow tat. Sic both ad z do ot dpd o ay patia coordiat thy cout with H ad z. Thrfor w ca add two or quatu ubr ad which rprt th agitud of pi aguar otu ad z-copot of pi aguar otu rpctivy. χ

3 Quatu Stat of Hydrog-ik Ato Th copt wavfuctio i ow th product of th patia wavfuctio Ψ ad th pi wavfuctio χ. Φ Ψ( r θ φ)χ W ow hav a t of FIVE quatu ubr ( ) to dcrib th quatu tat of a ctro i a hydrog-ik ato. - I ordr to ditiguih btw th quatu ubr rprtig z a z w add ubcript ad to th quatu ubr dcribig z ad z rpctivy. - So w hav for z ad for z. Bra-kt otatio: Sic w ca u th quatu ubr to pcify a quatu tat w o ogr d to xpicity writ out th wavfuctio thv. Itad w jut writ out th quatu ubr for th giv tat i uch a forat a or. i cad bra ad i cad kt. - - Bra ad kt ar copx cojugat to ach othr. - It ca ao b ud to xpr ir product * Φ Φ dr 3 3

4 Quatu Stat of Hydrog-ik Ato W ca ow rwrit th igvau quatio uig bra-kt otatio: ( ) ( ) z z o Z E E H ε π Lt u ow coidr th dgracy i a hydrog-ik ato. - Th rgy dpd oy o th pricipa quatu ubr. - For ach thr ar diffrt aowd vau of. : For ach pair of ad thr ar (+1) diffrt vau aowd for. : (-1) (-) -(-1) -. - For ach ad thr ar two aowd vau +½ or ½. - Thrfor thr ar a tota of (+1) tat for ach rgy v pcifid by. Thi i quivat to ay that thr i a (+1)-diioa vctor pac for ach t of () vau ad th (+1) fuctio for a orthoora bai t. 4

5 Ergy Lv of Hydrog-ik Ato Not tat 0 p tat 1 d tat Spctrocopic otatio: 1 p 3d tc. 5

6 Appicatio of Group Thory to Hydrog-ik Ato I tr of group thory th hydrog-ik ato i aid to po th fu rotatio ytry. That i it i ivariat udr a rotatio i 3-diioa pac. Thrfor th quatu tat of hydrog-ik ato ca b rprtd by th irrducib rprtatio of th fu rotatio group ad thir bai fuctio (or row ubr). It tur out that w ca cotruct irrducib rprtatio of th fu rotatio group uig th phrica haroic Y a th bai fuctio. - For a giv vau of thr ar (+1) Y fuctio with diffrt vau of. - Th fuctio for a bai t for a (+1)-diioa irrducib rprtatio. - Thrfor th aguar otu quatu ubr ca b rgardd a a idx to irrducib rprtatio which corrpod a ditict (+1)-fod dgrat rgy v. - Each vau of pcifi th bai fuctio or th row ubr withi th irrducib rprtatio. - Howvr th tat with diffrt but th a hav th a rgy. accidta dgracy. - Th group of a rotatio i 3D pac i a ifiit group uuay dotd by SO(3). 6

7 Sig-Ectro Cryta Fid Thory Suppo th ato (or io) i ow pacd iid a cryta. Th fu rotatio ytry i th rducd to a poit group ytry of th cryta. Lt u coidr a our firt xap a io cotaiig a ig ctro i th p tat i th outrot h. If thr i o xtra fid th ctroic tat ca b w dcribd by th hydrog-ik ato od. p tat corrpod to 1. - Spctrocopic otatio covtio: : 0 p : 1 d : f : 3 Th p tat cotai thr dgrat tat (or orbita) corrpodig to Th radia part of th wavfuctio i proportioa to r. - Th aguar part of th wavfuctio ar: Y π iφ i θ Y 10 3 coθ 4π Y π - W ca th cotruct 3 wavfuctio i th for of f(r)x f(r)y ad f(r)z. iφ i θ - Th orbita ar oft rfrrd to a p x p y ad p z bcau of th hap of thir wavfuctio. 7

8 Sig-Ectro Cryta Fid Thory Th ctro ca occupy ay of th 3 avaiab orbita i th p-h which ar dgrat. Suppo thi io i pacd i a cryta i which it i urroudd by 6 octahdray coordiatd io (.g. i a ip cubic tructur). Th ctroic wavfuctio ad thir rgy v wi b chagd du to th cryta fid. Howvr by ituitio w ca that a thr p-orbita ar affctd by th cryta fid i th a way a how bow. I othr word th cryta ytry do ot ditiguih th thr orbita of th p tat. Thy wi rai dgrat. 8

9 Sig p-ectro i a Cryta Fid W ca u th group thory to cofir our ituitio. I tr of ytry th fr io po fu rotatioa ytry. I th octahdra cryta th fu rotatioa ytry i rducd to th octahdra ytry O h. To do thi w u th rductio forua obtaid i Ca 6. a i 1 h R χ i) * ( i) ( R) χ ( R) N χ( C ) χ ( C ) ( 1 I ordr to u th rductio forua w d to kow th charactr χ(c k ) ad χ (i) (C k ) of th fu rotatio group ad th octahdra group. Not that C k ar th ca of th octahdra group. W kow th charactr of th fu octahdra group O h. - Th charactr tab for O i avaiab i th itratur. h - Th fu octahdra group O h O i - Thrfor th charactr tab for O h i: E 8C 3 3C 6C' 6C 4 i 8iC 3 3iC 6iC' 6iC 4 A 1g A g E g T 1g T g A 1u A u E u T 1u T u k k k k * 9

10 Fu Rotatio Group I ordr to fid th charactr χ(c k ) for th fu rotatio group w brify dicu th irrducib rprtatio of th fu rotatio group. Th fu rotatio group i obviouy a ifiit group. - Rotatio by th a ag (about whatvr ax) for a ca. - Thr ar ifiit ca ach of which cotai ifiit ubr of rotatio. - Thrfor thr ar ifiit irrducib rprtatio. W ca cotruct a (+1)-diioa irrducib rprtatio by uig th phrica haroic Y (θφ) a bai fuctio. - Y (θφ) i ao th wavfuctio of hydrog-ik ato. - Thr ar (+1) fuctio for giv. - Each rprt a irrducib rprtatio. It i o accidt that th outio of hydrog-ik ato i ao foud to b th bai fuctio of th irrducib rprtatio of th fu rotatio group. - Hydrog-ik ato po fu rotatio ytry. - Thrfor ach rgy v of th hydrog-ik ato ca b rprtd by a irrducib rprtatio of th fu rotatio group. 10

11 Fu Rotatio Group For th fu rotatio group - Th aguar otu quatu ubr tur out to b th irrducib rprtatio idx. - Each irrducib rprtatio D () i (+1)-diioa ad thu ha (+1) orthoora bai fuctio. - Each of th bai fuctio corrpod to diffrc vau. (For ow w oit th ubcript ). - Th dgracy btw th tat with diffrt vau but th a i accidta dgracy. Lt ow try to fid th charactr χ(c k ) of th irrducib rprtatio D (). - Sic a rotatio by th a ag for a ca w oy coidr th rotatio about th z-axi. - Appy rotatio P α by a ag α about z-axi o Y (θφ) P ( θ φ) ( θ φ α) α Y Y Rca th dfiitio of phrica haroic fuctio; Y ( + 1)( )! 4π( + )! iφ ( θ φ) ( 1) P ( θ) whr P (θ) i th aociat Lgdr poyoia. Thrfor w ow hav P iα ( θ φ) Y ( θ φ α) Y ( θ φ) α Y 11

12 Fu Rotatio Group W ow cotruct th (+1)-diioa rprtatio atrix D () (α) for th rotatio P α about th z-axi uig th Y (θφ) bai fuctio. - Each row ad cou of th atrix i abd by th (+1) aowd vau of. iα - Matrix t Y Pα Y δ. - D ( ) ( α) i 0 α - Th charactr i - For p tat χ ( ) ( α) i( 1) α ( 1) χ ( α) iα 0 iα i i i i ( + 1/ ) ( α / ) ( 3α / ) ( α / ) α 1

13 Fu Rotatio Group Sic th fu octahdra group O h ao cotai th ivrio opratio i additio to th ip rotatio w d to fid charactr χ (1) for ivrio ad ivrio-rotatio opratio. Firt t u cotruct th atrix for ivrio. Y - Appy ivrio P i o Y (θφ) P Y i ( θ φ) Y ( π θ φ + π) + P 1 ) - U th kow proprty of P (θ): ( π θ) ( ) P (θ - W th fid i( φ+π) + iφ ( π θ φ + π) P ( π θ) ( 1) P ( θ) ( 1) Y ( θ φ) Y P Y δ 1. - Matrix t i ( ) - D ( ) () i ( 1) 0 - Th charactr i - For th p v: χ ( ) ( 1) 0 ( 1) () i ( 1) ( 1)( + 1) ( 1) χ ( i) ( 1) 1 ( 1+ 1) 3 13

14 Fu Rotatio Group Fiay w arch for th charactr for ivrio-rotatio opratio. Appy ivrio P i ad rotatio P α qutiay o Y (θφ) P P Y i ( θ φ) Y ( π θ φ + π α) α Not th ordr of opratio i ot iportat ad ao w ca choo z-axi a th rotatio axi a bfor. + Oc agai w u ( π θ) ( ) P ( θ) Y P 1 to fid: ( π θ φ + π α) P ( π θ) i( φ+π α) + iα iφ iα ( 1) P ( θ) ( 1) Y ( θ φ) - Matrix t i α ( ) - D ( ) ( iα) ( 1) - Th charactr i χ ( ) - For th p v: 0 iα Y PP Y δ iα ( 1) 1. i( 1) α iα ( iα) ( 1) ( 1) ( ) ( 1) χ iα i i i i ( 3α / ) ( α / ) 0 ( ) iα 1 ( + 1/ ) ( α / ) α 14