Chapter 3 Basic Crystallography and Electron Diffraction from Crystals. Lecture 11. Chapter 3, CHEM 793, 2011 Fall, L. Ma
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1 Cpte 3 Bsic Cystopy nd Eecton Diffction fom Cysts Lectue
2 Pof. Sectmn: Nobe impossibe witout micoscope Isei ecipient of 0 cemisty Nobe Pize sys oundbein discoey of 'qusicysts' woud e been deyed fo yes witout powefu eecton micoscope. (ttp://
3 Ho-M-Zn dodeced cyst, own by usin te sef-fux metod (excess M), nd sowy cooin fom 700 C to 480 C. Te R-M- Zn fmiy is te fist e-et continin qusicyst stuctue, wic ows te study of ocized mnetic moments in qusipeiodic enionment. Eecton diffction ptten of n icosed Zn-M-Ho qusicyst
4 Atomic mode of A-A qusicyst
5 Top of tin foi Cyst pne () Bottom of tin foi B Lw d sinθ nλ
6 Equtions connectin te Cyst Pmetes (,, ) nd d-spcin wit bem pmetes (λ) ( ) Pne B Lw d (nm) d (nm) nλ d sinθ d d Inteye spcin of Atoms n. Tis is so Index λ Weent in nm Lttice pmete (nm) ( ) Cyst Pne o Mie Indices
7 Intoduction to te ecipoc ttice simpy in ecipoc ttice, sets of pe () e epesented by sine point octed distnce /d fom te ttice oiin insted of Mie index () tis definition is fom B s Lw, weein te ecto K is ecipocy eted to diffction(ө,λ) nd cyst d d sin λ sin θ θ d n n λ K
8 Exmpe Simpy, if sometin ( n object o ent) is e in e spce, ten it s sm in ecipoc spce
9 Mtemtic definition of te ecipoc ttice b c wee bcbbccbc0, i.e. is nom to bot b nd c,, etc bbcc, ony fo cubic, b, c e pe to, b,c Just s,b, nd c need not be nom one note,,b, nd c so not necessiy nom one note.
10 Popety of ecipoc ttice ecto. Zone w b c d If fmiy pne {} ies in zone [uw], i.e. pnes {} e pe to uw, ten is pependicu to uw, i.e. uw 0 uw ( u b w c ) ( b c ) so u u b w c w [uw] 0 Q (333) 0, b 0 () (uw) () A section of ecipoc ttice on (uw)
11 Popety of ecipoc ttice ecto. Zone w wit i ode pne b c d Conside te nt pne wic ies t pependicu distnce nd fom oiin. So te d-spcin nt nd uw [uw], ttice point on nt pne fom oiin Note: d nd n, n, n 0 O d uw [uw], ttice point on 0 pne tou oiin
12 Popety of ecipoc ttice ecto. Zone w wit i ode pne uw i nd nd i d ten uw i uw d so uw n substituti n fo uw nd ( u b w c ) ( b c ) Q, b 0 nd n so u w n
13 3. Zone xis t intesection of pne ( ) nd ( ) b c d If () nd () beon to zone [uw], ten we cn find te zone xis [uw], i.e. te diection of intesection of two pnes () nd () [uw] (333) u w 0 o u w 0 u w 0 () u u: : w ; : ( ) ; w : () A section of ecipoc ttice on (uw)
14 c b d 4. A pne () continin two Zone xes [u w ]nd [u w ] [uw] () ( ) ; ; : : : : u u u w u w w w u u w u w u w w w u w u o w u w u [uw]
15 5. d-spcin of ttice pne () d d b c d b c Q fo cubic cyst, b ( b d b b c c c ) ( c b c ) d
16 cos c b c b ρ d 6. Ane ρ between pne noms ( ) nd ( )
17 ) )( )( (.. e 7. Te ddition ue
18 c b b V b c c b c V c b c b c b V c b 8. Gene definition of,b,c in tem of, b, c Recipoc Lttice of Cysts: Re SC s SC nement of points e FCC s BCC nement of points e BCC s FCC nement of points (000) (00) (00) (00) (0) (0) () ()
19 Mtemtic epesenttion of ecipoc ttice We wnt ecipoc ttice ectos suc tt te ecipoc ecto is te inese in mnitude of te e ecto nd is nom to te pnes septin te oiin ecto. In ene Fouie Anysis of Peiodic Potenti Te peiodic potenti of ttice is ien by: wee U is te coefficient of te potenti, nd is e position ecto Howee ony ues of K e owed wic e ecipoc ttice ectos (S).
20 Poof: since U() U( R), wee R is ttice ecto, Let λ exp(iπ SR) nd S R n, wee n is n intee. Ony possibe ues e of te fom: G b c s GR nd,, e intees. Note: Tis is sticty te cystope s definition of ecipoc ttice ectos.
21 0. Exmpe: Setc to sce te () ecipoc pne fo body cente cubic (--) (-0)
22 Summy of Recipoc Lttice Te ecipoc ttice is so ced becuse ents e in ecipoc units. Te ecipoc ttice ies us metod fo pictuin te eomety of diffction. Geometicy te diffction ptten is te section of ecipoc pne fom ecipoc ttice on te zone xis
23 HW#9: Index te sdow cyst pnes wit Mie Indices. Due dy: Oct. 7 O O O O (00) (0) (-0) (-0)
Chapter 3 Basic Crystallography and Electron Diffraction from Crystals. Lecture 9. CHEM 793, 2008 Fall
Cpte 3 Bsic Cystopy nd Eecton Diffction fom Cysts Lectue 9 Top of tin foi Cyst pne () Bottom of tin foi B Lw d sinθ n Equtions connectin te Cyst metes (,, ) nd d-spcin wit bem pmetes () ( ) ne B Lw d (nm)
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