Chapter 3 Basic Crystallography and Electron Diffraction from Crystals. Lecture 12. CHEM 793, 2008 Fall

Transcription

1 Chapt 3 Bac Cytalloaphy and Elcton Dacton om Cytal Lctu 1 CHEM 793, 008 all

2 Announcmnt Mdtm Exam: Oct., Wdnday, :30 4:30 CHEM 793, 008 all

3 Th xctaton o, Ba' Law and th Lau quaton pdct dacton at only pc Ba anl o an nnt cytal. Many dacton xpmnt (pcally n TEM) a cad out on pcmn whch a thn n at lat on dmnon. Th ct o mall dmnon to allow dacton ov a an o anl clo to th Ba anl. Th ha th am ct a th latv cpocal lattc pont (lpont a hown n u (a) ) w ttchd out n th dcton o thnn o th ampl. Th tad cpocal lattc pont a now calld lod (u b). Why w tll dacton, whn th Ba condton not xactly atd? a b b l-pont l-od CHEM 793, 008 all

4 Th xctaton o Th dvaton paamt, a Th Ewald ph can ntct wth a lod vn whn t m th actual cpocal lattc pont. Dacton, at ducd ntnty, can thn tll occu. Th dvaton paamt,, dn how clo a patcula lod to th Ewald ph. I w allow tan, th dacton vcto K thn vn by vctoally addn th dvaton paamt to th cpocal vcto, o: K b Th dvaton paamt dnd to b potv n th dcton o th bam (downwad ) and natv t pont upwad a hown n u b. Th vcto,, a mau o how a w dvat om th xact Ba condton. CHEM 793, 008 all

5 Th xctaton o Th dvaton paamt, a In cpocal pac, th dacton vcto K vn by: K D I A dactd bam only a whn K.. t a vcto btwn cpocal lattc pont. I w allow tan o cpocal lattc pont, thn th dacton vcto vn by: K K In a thn cytal, dacton may b thu b n om a patcula t o ncdnt bam anl clo toth (not jut a nl anl), and/o a an o cytal ontaton. Th ct o tan that lattc pont whch do not touch Ewald' ph but a clo, can tll v dactd bam. Howv, thy wll hav a ducd bam ntnty. Th ntnty o th dactd bam va wth th valu o th dvaton paamt a hown n u b CHEM 793, 008 all b

6 Knmatcal Thoy o Elcton dacton Dcbn th anula dpndnc o th dactd wav, ψ(k), mttd om dnt aanmnt o atom. Explann how a tanlatonally-podc aanmnt o atom n a cytal pmt ton contuctv ntnc btwn ndvdual wavlt, catn th Ba dacton. Aumpton o nmatcal thoy that th ncdnt wav cattd latcally cohntly by ndvdual atom. Knmatcal thoy can b ud to calculat th tuctu acto o th unt cll. o lcton dacton contat om la atu uch a cytal hap and cytalln dct, nmatcal thoy uually qualtatv. Knmatcal thoy mo quanttatv o X-ay dacton bcau X-ay cattn much wa than lcton dacton. Quanttatv ult o ton lcton dacton qu th dynamcal thoy, whch wll not b dcud n dtal n th cla. Chc th txtboo o mo nomaton about dynamc thoy CHEM 793, 008 all

7 Elcton dacton om a matal h 8m 0 ψ ( ') [ E V ( ')] ψ ( ') 0 Dtcto K Th ncdnt lcton wav nd th cattn atom at th tmndpndnt SchÖdn quaton. h: Pan contant, m 0 : tatonay lcton ma, : atom coodnat E: potntal o lcton V: potntal o matal K K o K0 K0 KK- K0 Wav-vcto and poton vcto o lcton cattn CHEM 793, 008 all

8 CHEM 793, 008 all 0 ') ( ')] ( [ ') ( 8 0 V E m h ψ ψ I th wav undmnhd and cattd only onc by atom ( th aumpton vald whn th cattn wa). W hav th t Bon appoxmaton oluton: ' 3 ' ' ) ( d V h m ψ So th cattd pat o th wav ' 3 ' ' 0 ) ( 0 d V h m catt ψ

9 CHEM 793, 008 all Th cattd pat o th wav ' 3 ' ' 0 ) ( 0 d V h m ψ () th cattn acto. Th cattd wav popotonal to th ou tanom o th cattn potntal. w wll mply th tm to apply t.

10 CHEM 793, 008 all S at ψ ψ acto tuctu acto hap S ba at lattc

11 So, bcc tuctu acto ul: Th um o th th nt h,, l mut b an vn numb. o xampl, bcc W, th lowt-od allowd dacton a (110), (00), (11), (0), (310), (), (31), (440), (330), (411), (40), tc., but dacton uch a (100), (111), (10), tc. a obddn. Th ul appl to th oth cntd lattc: body cntd, thohombc, and body cntd ttaonal. obddn Dacton CHEM 793, 008 all

12 Only al pat dtmn th tuctu acto n blow quaton, th tm K. nt o cytal ba at tuctu acto h a* b* l c* x a y b z c a vcto whch dn th locaton o ach atom wthn unt cll, thn w can wt tuctu acto a: hl ( hx y lz ) o am atom n unt cll, ha dntcal valu CHEM 793, 008 all

13 hl ( hx y lz ) o bcc cytal: th lattc pont nclud (0,0,0), (1/,1/,1/) bcc bcc bcc ( hx y lz ) { ( h l ) 1 } h l vn 0 h l odd o cc acto ul: th th nt h,,l mut b all vn o all odd. o xampl, th lowt od dacton a (111), (00), (0), (311), (), (400), (331), (40), but oth dacton uch a th (100), (110), (10), (11), tc. a obddn. CHEM 793, 008 all

14 Suplattc Dacton Appln th bcc analy to B tuctu uch a NAl ntmtallc o B tuctu, th atom n th cnt dnt om atom at con. o NAl, Al n cnt, and N n con. So, N ba (0,0,0), and Al ba (1/,1/,1/) Thn NAl NAl NAl N N N N Al Al ( hx y lz ) ( h0 0 l0) Al ( h l ) Al h l vn h l odd h l Intad o zo dactd ntnty, th (100) dactd om B-odd NAl ha an ntnty popotonal to: I(100) N Al wa CHEM 793, 008 all Al N B-NAl unt cll

15 (00) NAl ( hx y lz ) (100) (00) NAl NAl N N N N Al Al ( h0 0 l0) Al ( h l ) Al h l vn h l odd h l NAl (001) dacton (010) On th oth hand, th allowd dacton om bcc cytal, th undamntal dacton,.. th (00), hav ntnty: I (00) N Al ton Th (100) dacton calld a uplattc dacton. It lct th podcty o c lattc upon whch B tuctu contuctd un a ba o two dnt atom. CHEM 793, 008 all Al N B-NAl unt cll

16 (00) (00) (100) (010) (00) (100) Wa ton (000) (010) (00) B-NAl up-lattc (001) dacton pattn CHEM 793, 008 all

17 Cytal Shap acto S lattc ba at hap acto tuctu acto o vy la cytal, th hap acto v lttl nomaton about th cytal hap, and not vy nttn,.. o c cytal: ψ I catt S c ψ ψ * N at c N at So o vy la cytal, th hap acto ntnty bcom nntly hh and nntmally naow. Th hap acto mot nttn o mall cytal. CHEM 793, 008 all

18 Cytal Shap acto S lattc hap acto o convnnc, w acc om nalty by aumn that th mall cytal a ctanula pm wth N x,n y, and N z unt cll alon th dcton a hown n u c N y b N x a N z c b a An thn-ol pcmn modld a a ctanula lab mad up o ctanula unt cll o d a,b,c. Th a Nx cll n th x dcton, Ny cll n th y dcton, and Nz n th z dcton Lt n n n z n x y n x a n 0,1... N 0,1... N 0,1... N x y z y b n z c CHEM 793, 008 all

19 CHEM 793, 008 all catt I maxmum th Lau condton and th ntnty a Th nt, whn n n n n n n, cattd wav Th ntnty o N a c c N b b N a a N I I z y x catt ψ ψ 3-D dtbuton o I z y x x N n K c N n K b N n K a n K a N,, n an nt,..., whn Th lad to th l-od da Th a ubday maxma o mnma o I a hown n u

20 CHEM 793, 008 all Dvaton vcto (): thn-ol ct c b a o c b a nt and notn that ba at lattc acto tuctu acto hap S and c o n b m a

21 CHEM 793, 008 all S S S lattc lattc lattc lattc lattc nt Th hap acto dpnd only on not

22 CHEM 793, 008 all ( ) ( ) l h l h at o n m at o n m at o n m l h at at a a at c b a c b a c b a ) ( ) 1 ( numb vn 0 ) 1 ( oddnumb cytal a bcc o ), ( ) (0, ), ( ) ( ) ( )/ ( )/ ( 0 0 tm ) ( (000), Th tuctu acto dpnd only on. Cond a bcc cytal

23 CHEM 793, 008 all c c N b b N a a N I I z y x catt ψ ψ catt n n n n n n, cattd wav Th ntnty o Th hap acto dpnd only on not. Th dactd ntnty not a contant o any poton alon th l-od. -D dtbuton o I I Th th nmatcal ntnty dtbuton about any cpocal lattc pont. Th ntnty popotonal to N whnv 0. I th cytal ha a unom thcn (no vaaton n Nx o xampl), th ntnty quaton how th a maxma and mnma n th dacton ntnty wth ncan valu o

24 ө Gomty dcpton o dacton ntnty vayn wth Obvd ntnty K S<0 S0 S>0 Ewald ph Rl-od Th l-od at hl whn th bam ө away om th xact Ba condton. Th Ewald ph ntcpt th lod at natv valu o whch dn th vcto K. Th ntnty o th dactd bam a a uncton o wh th Ewald ph cut th l-od hown on th ht o th daam. In th CHEM ca, 793, th 008 ntnty all ha alln almot zo.

25 Applcaton o hap acto, S () 1. Dacton om wd-hapd pcmn Top uac u how dacton om a wdd cytal. Rl-od alway nomal to th uac. Ewadl Sph Bottom uac whn >0, th two lod (l-1 and l-) nat a doublt. Rl-1 pot on th lt. Th mddl th matx l-od. G 1 l-1 1 S>0 l- G CHEM 793, 008 all

26 Applcaton o hap acto 1. Dacton om wd-hapd pcmn Top uac u how dacton om a wdd cytal. Rl-od alway nomal to th uac. Ewald Sph G S<0 Bottom uac whn <0, th two lod (l-1 and l-) nat a doublt. Rl-1 pot on th ht. Th mddl th matx l-od. 1 1 G S<0 l-1 l- CHEM 793, 008 all

27 Applcaton o hap acto 1. Dacton om wd-hapd pcmn Snc th lnth o l-od nvly popotonal to th pcmn thcn, th thnn th pcmn, th mo dacton pot wll occu n th pattn. A hown n u, a o dacton pattn w tan om a wdd cytal ampl. Th numb o dacton pot dca a th thcn o pcmn nca CHEM 793, 008 all

28 Applcaton o hap acto. Dacton om plana dct Spcmn Th ct o a thn nclnd plat n a ol. Whn 0, and th no plat, w only on pot n DP (dacton pattn) bcau th on l-od. O G Ewald Sph m: l-od o uac S0, and no platlt nd pcmn Inclnd platlt O G Spcmn CHEM 793, 008 all

29 Applcaton o hap acto. Dacton om plana dct Inclnd platlt Spcmn Th ct o a thn nclnd plat n a ol. N G Ewald Sph Whn 0, and th no plat, w only on pot n DP (dacton pattn) bcau th on l-od. O n: l-od o plat m: l-od o uac Whn <0, and th a plat, w two pot bcau th a two lod O N G S<0, and plat CHEM 793, 008 all

30 Applcaton o hap acto 3. Dacton om patcl Th pncpl n dtmnn th hap acto ntnty dtbuton o vaou patcl o cytal a mply mall bcom la and vc va. Th l-od hap alo appoxmatly th lct th dacton ntnty dtbuton. Small bcom la CHEM 793, 008 all La bcom mall

31 Applcaton o hap acto 3. Dacton om patcl Exampl o how pot n cpocal pac hav dnt hap, dpndn on th patcl whch a dactn z x y Th pncpl n dtmnn th hap acto ntnty dtbuton o vaou patcl o cytal a mply mall bcom la and vc va. Th l-od hap alo appoxmatly th lct th dacton ntnty dtbuton. Du cattn CHEM 793, 008 all

32 Applcaton o hap acto 4. Dacton om thn cytalln plat whch paalll to th lcton bam (a) (b) (a) Schmatc pntaton o th pcmn hown th pcptat, thn cytalln plat, a paalll to th lcton bam. (b). Th copondn dacton pattn how th lon ta( du cattn) caud by th thn plat-l patcl Sta CHEM 793, 008 all

33 Applcaton o hap acto 5. Dacton om nly twn tuctu whch nomal to th lcton bam Sta Twn tuctu (b). a how nly twnnd matntc Co NGa ol. Th twn tuctu dvd th cytal pac nto paat on wth vayn wdth, and thby wll v to om ot o ta n dacton pattn. Th dcton o ta, o cou, ppndcula to th twn plan.. b how th copondn zon ax dacton pattn. Th ta a nomal to th twn plan. Each cpocal lattc pont ha two pa o modulaton atllt, whch alo v to ta. CHEM 793, 008 all

34 HW#13: A pcptat ha th hap o ttahdon wth lnth o a a hown blow. What hould b th hap o th du cattn aound th undamntal dacton (.. lod hap)? Stch t a pcly a pobl, and ndcat th lnth n cpocal pac copondn to AB n al-pac. Not that ABBCACOCa Du: Oct. 15/08 C O A a B CHEM 793, 008 all