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) d (nm) n d sinθ d d Inteye spcin of Atoms n. Tis is so Index Weent in nm Lttice pmete (nm) ( ) Cyst ne o Mie Indices
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 is ecipocy eted to diffction(ө,) nd cyst d d sin sin θ θ d n n
Exmpe Simpy, if sometin ( n object o ent) is e in e spce, ten it s sm in ecipoc spce
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.
opety 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) 333 000 () A section of ecipoc ttice on (uw)
opety 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
opety 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
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 : 333 000 () A section of ecipoc ttice on (uw)
c b d 4. A pne () continin two Zone xes [u w ]nd [u w ] [uw] () ( ) ; ; : : : : 0 0 0 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]
c b d 5. d-spcin of ttice pne () fo cubic cyst, ) ( ) ( d c b c b d c c b b d c b c b d Q
cos c b c b ρ d 6. Ane ρ between pne noms ( ) nd ( )
0 00 0 ) )( )( (.. e 7. Te ddition ue
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) () ()
0. Exmpe: Setc to sce te () ecipoc pne fo body cente cubic 0-4 --3 -- -3- -40 (--) (-0) -0-0 000-0 -0 40-3- - 3-04-
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
HW#9: Index te sdow cyst pnes wit Mie Indices. Due dy: Oct. 06/08 O O O O (00) (0) (-0) (-0) HW# 0: Setc to sce te () ecipoc pne fo fce cente cubic Due Oct. 06/08
Top of tin foi Cyst pne () Bottom of tin foi B Lw d sinθ n Diffction pocess in tems of B Lw by cyst pne: d-spcin
Lue Condition: diffction pocess in tems of scttein by indiidu toms As sown in Fiue (b), diffction occus wen is ecto te ecipoc ttice ( ), i.e. D - I Incident we ecto ө ө Diffcted we ecto I D I D () Refection t te B ne θ fom cyst pnes (b) Te ecto dim (ecto tine) descibin te diffction pocess
Lue Condition: diffction pocess in tems of scttein by indiidu toms. Fisty, we wi use ecto eomety to cec Lue condition Incident we Te sctteed wes wi be in pse if te pt diffeence,, is n inte numbe of weents. In ecto nottion: Atom position O L O 0 OD, nd OL OD O 0 0 D Lttice ecto OL 0: Unit ecto of incident we Diffcted we : Unit ecto of sctteed we Te sctteed wes wi be in pse, if 0 n
0 0 n o n Te sctteed wes wi be in pse, if Usin to nomize te boe eqution,,,, / / / 0 0 0 nd note so nd et D I I D D I I D ө ө, I nd D my be descibed in tems of ecto tine s sown beow
n o n so n n D I 0 0, I D ө ө, I nd D my be descibed in tems of ecto tine s sown beow Lue Condition Te tom position ecto, ttice ecto, times te ecto diffeence of incident we, I nd diffction we D is intee Unde tis condition, te incident nd diffcted we is in pse
Lue Condition As sown in Fiue, usin ecto tine to cec Lue condition et nd I D in ecto tine on te it d I sinθ, ten sinθ i. e. d d sinθ, I nd D my be descibed in tems of ecto tine s sown beow I ө ө D Tis is B Lw
Lue Condition. Secondy, we cn use te ecto eb to cec Lue condition In ecipoc spce In e spce uw b u b w c c, I nd D my be descibed in tems of ecto tine s sown beow I ө ө D
Lue Condition In ecipoc et uw b c u b w c In e spce uw b c u b w c u w n spce, I nd D my be descibed in tems of ecto tine s sown beow I ө ө D Tis is equient to Lue condition eqution: n
Lue Condition A speci cse is settin equ to te tee unit ectos in tun, te sctteed wes wi be in pse if Diffction on teedimension ttice b c
Summy of Lue Condition Diffction occus wen te ecto of diffction we is ecto te ecipoc ttice, ), i.e.. Tis condition is equient to B Lw. B s Lw ctuy is mtemticy coect but pysicy won, since it doesn t conside 3-D of cyst Lue condition descibes te cyst diffction in ioous fsion, I nd D my be descibed in tems of ecto tine s sown beow Next ectue: Ewd spee nd stuctue fcto ө ө I D