Chapter 3 Basic Crystallography and Electron Diffraction from Crystals. Lecture 11. CHEM 793, 2008 Fall

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1 Chapt 3 Basc Cystalloaphy and Elcton Dacton om Cystals Lctu CHEM all

2 Top o thn ol Cystal plan (hl) Bottom o thn ol Ba Law d snθ nλ hl CHEM all

3 Equons connctn th Cystal Paamts (h l) and d-spacn wth bam paamts (λ) (h l) Plan Ba Law d (nm) d (nm) nλ d snθ d hl hl h a l CHEM all d Intlay spacn o Atoms n. Ths s also Indx λ Wavlnth n nm a Ltc paamt (nm) (h l) Cystal Plan o Mll Indcs

4 Mhmcal dnton o th cpocal ltc v hl ha * b * lc * d hl hl wh a* ba*cb*ab*cc*bc*a.. a* s nomal to both b and c tc a*ab*bc*c only o cubc a* b* c* a paalll to a bc Just as ab and c nd not b nomal on anoth a*b* and c* also not ncssaly nomal on anoth. CHEM all

5 Lau Condton: dacton pocss n tms o sctn by ndvdual oms As shown n u (b) dacton occus whn K s a vcto th cpocal ltc ( hl ).. KK D -K I hl Incdnt wav vcto ө ө Dactd wav vcto K I K D K I K D (a) Rlcton th Ba anl θ om cystal plans CHEM all K (b) Th vcto daam (vcto tanl) dscbn th dacton pocss and K K hl I K K D λ hl d hl

6 Constucton o th Ewald s sph cont (c) Constuct a ccl wth adus /λ (.. I ) whch passs thouh (c). Whv a l-pont touchs th ccl Ba's Law s obyd and a dactd bam wll occu (d). CO psnts th ncdnt bam and CG s a dactd bam. Th anl btwn thm must b Θ. OG s th hl vcto and thus has mantud /d hl and snc /λ. OG (/ λ)sn Θ /d hl aann ths vs Ba's quon wth n : nλ dsn Θ CHEM all (d)

7 Th xcton o s Ba's Law and th Lau quons pdct dacton only pcs Ba anls o an nnt cystal. Many dacton xpmnts (spcally n TEM) a cad out on spcmns whch a thn n last on dmnson. Th ct o small dmnsons s to allow dacton ov a an o anls clos to th Ba anl. Ths has th sam ct as th lv cpocal ltc ponts (lponts as shown n u (a) ) w sttchd out n th dcton o thnnss o th sampl. Th stad cpocal ltc ponts a now calld lods (u b). Why w stll s dacton whn th Ba s condton s not xactly ssd? a b b l-pont l-od CHEM all

8 Th xcton o Th dvon paamt s a Th Ewald sph can ntsct wth a lod vn whn t msss th actual cpocal ltc pont. Dacton ducd ntnsty can thn stll occu. Th dvon paamt s dns how clos a pcula lod s to th Ewald sph. I w allow stan th dacton vcto K s thn vn by vctoally addn th dvon paamt s to th cpocal vcto so: K s b Th dvon paamt s dnd to b postv n th dcton o th bam (downwads ) and nv t ponts upwads as shown n u b. Th vcto s s a masu o how a w dv om th xact Ba condton. CHEM all

9 Th xcton o Th dvon paamt s a In cpocal spac th dacton vcto K s vn by: K D I A dactd bam only ass whn K.. t s a vcto btwn cpocal ltc ponts. I w allow stan o cpocal ltc ponts thn th dacton vcto s vn by: K s In a thn cystal dacton may b thus b sn om a pcula st o ncdnt bam anls clos toth (not just a snl anl) and/o a an o cystal ontons. Th ct o stan s th ltc ponts whch do not touch Ewald's sph but a clos can stll v dactd bams. Howv thy wll hav a ducd bam ntnsty. Th ntnsty o th dactd bam vas wth th valu o th dvon paamt s as shown n u b CHEM all b

10 Knmcal Thoy o Elcton dacton Dscbn th anula dpndnc o th dactd wav ψ(k) mttd om dnt aanmnts o oms. Explann how a tanslonally-podc aanmnt o om n a cystal pmts ston constuctv ntnc btwn ndvdual wavlts cn th Ba dactons. Assumpton o nmcal thoy s th th ncdnt wav s sctd lastcally cohntly by ndvdual oms. Knmcal thoy can b usd to calcul th stuctu acto o th unt cll. o lcton dacton contast om la us such as cystal shaps and cystalln dcts nmcal thoy s usually qualtv. Knmcal thoy s mo quanttv o X-ay dacton bcaus X-ay sctn s much wa than lcton dacton. Quanttv sults o ston lcton dacton qu th dynamcal thoy whch wll not b dscussd n dtal n ths class. Chc th txtboo o mo nomon about dynamc thoy CHEM all

11 Intnsty o dacton Lann Objctvs: At compltn ths scton you should b abl to: xplan why non-pmtv unt clls v s to vaons n th ntnsty o dacton spots om dnt plans; pdct whch obddn lctons o any non-pmtv cystal stuctu; and wt down th obddn lctons o basc stuctus. CHEM all

12 Intnsty o dacton In stuctus wth non-pmtv clls sctn om on om n th cll can nt wth sctn om anoth to duc o ncas th ntnsty o dacton. Dnt dacton ntnsty and dacton ptns du to dnt stuctus o spcmns CHEM all

13 Elcton dacton om a mal h 8m ψ ( ') [ E V ( ')] ψ ( ') Dtcto K Th ncdnt lcton wav nsd th sctn om sss th tmndpndnt SchÖdn quon. h: Pan constant m : stonay lcton mass : om coodn E: potntal o lcton V: potntal o mal K K o K K KK- K Wav-vctos and poston vctos o lcton sctn CHEM all

14 CHEM all ') ( ')] ( [ ') ( 8 V E m h ψ ψ I th wav s undmnshd and sctd only onc by om ( ths assumpton s vald whn th sctn s wa). W hav th st Bon appoxmon soluton: ' 3 ' ' ) ( d V h m ψ So th sctd p o th wav s ' 3 ' ' ) ( d V h m sct ψ

15 CHEM all Th sctd p o th wav ' 3 ' ' ) ( d V h m ψ () s th sctn acto. Th sctd wav s popotonal to th ou tansom o th sctn potntal. w wll smply ths tm to apply t.

16 CHEM all R j j sct R V V d V h m ' ' ) ( ' 3 ' ' ψ Rj Rn - Rn Not th whnv Rj on o tms o V s V () and th potntal V ( ) ts a ston contbuton om om cntd Rj Dn a nw coodn: - R j Tho w can chan th xponntals om ull phas actos o ndpndnt wavlts nto lv phas actos o wavlts om th dnt oms substtut th nw coodn nto abov quon

17 CHEM all d V h m j j j R R R 3 ψ j j j R R R d V h m ψ 3 Dn snl omc acto o lcton sctn d V h m R R l j 3 Only consd sctd wav nsd th mal (cystal) lav out th acto

18 Consdn K s vy small w t l as a numb only dpndn on th typ o om locd R j. So th sctd wav om N oms s wttn most smply as ψ N j l ( R ) j R j Th dactd wav s popotonal to th ou tansoms o th sctn acto dstbuton n th mal CHEM all

19 Dacton om a ltc wth a bass In both al spac and n cpocal spac t s usul to dvd a cystal composd o oms locons nto ps accodn to pscpton: Cystal ltcbass dct dsplacmnts δ Th ltc s on o th 4 Bavas ltc typ. Th cystal typcally has numous unt clls on ts ltc and numous Th bass s th om oup assocd wth ach ltc st th unt cll typcally has a w o xampl bcc stuctu { } {() (///)} { } {mnl} any nts o a dct- cystal th om postons R a povdd by vcto to ach unt cll R CHEM all

20 CHEM all Dacton om a ltc wth a bass R R R R ψ ψ So th sctd wav o th cas o an nntly la dct- ltc wth a bass Th om bass s dntcal o all unt clls ( ) ( )

21 CHEM all S ψ ψ acto stuctu acto shap S bass ltc

22 CHEM all Snc th stuctu acto o unt cll s dntcal o all ltc ponts t s usually convnnt to wt th dactd wav as: ltc ψ Th ntnsty o th dactd wav dtcto I s th poduct o th wavuncton wth ts complx conju: )* ( ) ( ) ( I d ψ ψ

23 CHEM all Stuctu acto uls. Stuctu acto o sc ltc o a sampl cubc (sc) ltc w can show asly th ston dacton occus o any nt combnon (hl). A nal sc cpocal ltc vcto * * * c l b h a o oms locd on th sts o a smpl cubc ltc: c b a and o c n b m a

24 Stuctu acto uls Accodn to Lau condton th ston dacton occus K ha* b* l c* hm n lo nt ma nb oc o any and nt hl ha* b* l c* CHEM all a b c

25 CHEM all Tho whn th Lau condton s ssd o th smpl cubc ltc: ( ) ( ) ( ) N S tm N N nt Th sctd wav s N S sc sc ψ o sc cystal any cystal (hl) poducs th ston dacton

26 CHEM all o bcc cystal Th a two bass vctos and : { } {() (///)} Th ltc s { } {mnl} any nts l h bcc tm bcc l h l h ) ( () Accodn to Lau condton th ston dacton occus K

27 CHEM all l h bcc ) ( () Th stuctu acto tas on two valus dpndn on whth th sum hl s odd o vn vn numb s ) ( () ) ( () odd numb s l h N ) ( () ) ( () l h N N bcc N bcc

28 So bcc stuctu acto ul: Th sum o th th nts h l must b an vn numb. o xampl bcc W th lowst-od allowd dactons a () () () () (3) () (3) (44) (33) (4) (4) tc. but dactons such as () () () tc. a obddn. Ths ul appls to th oth cntd ltc: body cntd thohombc and body cntd ttaonal. obddn Dactons CHEM all

29 Only al p dtmns th stuctu acto n blow quon th tm K. s nt o cystal bass stuctu acto h a* b* l c* x a y b z c s a vcto whch dns th locon o ach om wthn unt cll thn w can wt stuctu acto as: hl ( hx y lz ) o sam om n unt cll has dntcal valu CHEM all

30 hl ( hx y lz ) o bcc cystal: th ltc ponts ncluds () (///) bcc bcc bcc ( hx y lz ) { ( h l ) } h l s vn h l s odd HW#: Pov th cc acto ul: th th nts hl must b all vn o all odd. o xampl th lowst od dactons a () () () (3) () (4) (33) (4) but oth dactons such as th () () () () tc. a obddn. Du day: /3/8 CHEM all

31 Supltc Dactons Appln th bcc analyss to B stuctu such as NAl ntmtallcs o B stuctu th om n th cnt s dnt om oms con. o NAl Al s n cnt and N s n con. So N bass s () and Al bass s (///) Thn NAl NAl NAl N N N N Al Al ( hx y lz ) ( h l) Al ( h l ) Al h l s vn h l s odd h l Instad o zo dactd ntnsty th () dactd om B-odd NAl has an ntnsty popotonal to: I() N Al wa CHEM all Al N B-NAl unt cll

32 () NAl ( hx y lz ) () () NAl NAl N N N N Al Al ( h l) Al ( h l ) Al h l s vn h l s odd h l NAl () dacton () On th oth hand th allowd dactons om bcc cystal th undamntal dactons.. th () hav ntnsty: I () N Al ston Th () dacton s calld a supltc dacton. It lcts th podcty o sc ltc upon whch B stuctu s constuctd usn a bass o two dnt oms. CHEM all Al N B-NAl unt cll

33 () () () () () () Wa ston () () () B-NAl sup-ltc () dacton ptn CHEM all

34 So to obtan th supltc dactons o an odd stuctu st loc th undamntal dacton o undlyn ltc (no th om typ).. N n abov xampl. Nxt locd th dactons om a modd ltc wh on spcs o oms s movd.. Al n abov xampl. Th unt cll s now la so th a mo dactons. Th supltc dacton occu th locons o th nw dactons o ths modd ltc. HW#: 3AlC phas n -C-A systm has a cubc stuctu: Al s con C s n th cubc cnt and s n th cnt o ach ac.. Dv an xpsson o th stuctu acto n tms o Al and C. Stch th ()* scton o th cpocal stuctu o ths 3AlC phas labln th low ndx dactons and ndcn lv ntnsts. Du: /3/8 C Al CHEM all

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