X-ray Diffraction beyond the Kinematical Approximation

Transcription

1 X-ay Diffaction beyond the Kinematical Appoximation Dynamical theoy of diffaction Inteaction of wave fields X-ays neutons electons with a egula lattice atomic cystal stuctues nanomete scaled multi-layes self aanged systems

2 Intoduction Bagg scatteing Ideally impefect cystal Pefect cystal

3 Intoduction Kinematical diffaction Dynamical diffaction Physics Macoscopically impefect cystals Scatteing fom micoscopic mosaic blocks of the cystal Magnitude of X-ays wave field assumed not to change ove these blocks Kinematical calculations Macoscopically pefect cystals Scatteing fom nealy pefect single cystals Magnitude of X-ay wave field changes significantly due to diffaction (not absoption!) Dynamical calculations Mathematics No inteaction between incident and diffacted waves consideed Only single scatteing events Fouie Tansfom Ewald constuction (Bagg s law ) Coheent supeposition of all diffacted waves Multiple scatteing events included Maxwell s Equations in peiodic media Modified Ewald constuction (dispesion sufaces )

4 Intoduction Kinematical diffaction Dynamical diffaction

5 Intoduction Kinematical diffaction Ewald sphee constuction Dynamical diffaction Ewald sphee constuction with efaction coection nλ= d sinθ B???

6 istoical Milestones - Theoy Dynamical Theoy C. G. Dawin 1914 Dawin Dynamical teatment of Bagg eflection 1916 Ewald Genealized dynamical teatment of (light) scatteing micoscopic theoy: 3D aay of dipoles 193 Pins Extension of Dawin theoy to absobing cystals 1931 Laue Dynamical calculation of cystal susceptibility macoscopic theoy: Maxwell-equation in cystals P. P. Ewald 1957 amilton Dynamical teatment of mosaic cystals 198 Kato Statistical desciption of defomed/mosaic cystals 1994 Caticha Dynamical diffaction theoy fo θ B π/ M. von Laue

7 Qualitative Desciption Mathematical teatment of dynamical theoy: Diffeential equations Solve Maxwell s equations in a medium of the peiodic dielectic constant (elated to the electon density of the lattice) Gaphic epesentation: Dispesion sufaces Modified Ewald sphee constuction

8 Fundamental Equations von Laues appoach: 1 E = ρ ε E B-εµ = t B = B E + = t Maxwell s equations peiodic medium single fequency Ewald-Bloch Ansatz 1 i ρ ( ) = Fe V iωt E(,t) = E( ) e ( ) ( E = E e ) i K + K Stuctue facto F F N = F + if = n= 1 [ f ( ) + f ( λ) + i f ( λ) ] exp( i ) D ( ) n n

9 ( ) ( ) 1 1 = Γ Γ Γ Γ E E K K F k F p k F p k K K F k ( ) ( ) ( ) ( ) F F p k K K F k K K F k Γ = Γ Γ Two beam diffaction Assumptions: Only two coefficients E and E contibute. This set of equations has only non-zeo solutions if the dispesion equation is fulfilled. K K V k e + = = Γ and 4 with π X-ay Diffaction beyond the Kinematical Appoximation Fundamental Equations

10 Fundamental Equations The dispesion suface The solutions of the dispesion equation define a suface in ecipocal space. Gap in the dispesion suface at the bode of the Billouin zone with no tavelling waves Evey point on this suface coesponds to a wavefield of plane waves with E E k ( 1 ΓF ) = k K pγf K

11 Symmetical Bagg eflection The Dawin Cuve E E FF = exp(i ν ) = η± η 1 E E F ( ) Nomalized angle paamete η η ΓF θsinθ Γ FF B θ = θ θ B : deviation fom the kinematical Bagg angle

12 The Dawin Cuve The Dawin width w = θ θ = η= 1 η=+ 1 Γ F F F F F sinθ + B angula width typically ~ ac seconds (But fo θ B 9 : w?!?) width w ~ stuctue facto F integated intensity I ~ stuctue facto F

13 The Dawin Cuve Refaction effect η ΓF θsinθ Γ FF B fo θ = : η!

14 The Dawin cuve Dawin width as function of E and θ B Fo given Bagg eflection width w inceases with θ B Fo highe Bagg eflections width w deceases (constant E) The extinction depth Λ ext Fo stong Bagg eflections Λ ext < Λ abs Fo weak Bagg eflections Λ ext > Λ abs

15 The Dawin Cuve Example: Bagg eflection on coppe (Z=9) lattice plane spacing d 111 =.87 Å Case 1: θ B < 9 E = 8.5 kev (λ = 1.54 Å) θ B = 1.65 f + f +i f = i.67 F = i.43 F 111 = i.43 Λ ext = Å E = 15.9 ev w = Case : θ B 9 E =.97 kev (λ = 4.17 Å) θ B = 88 f + f +i f = i 3.4 F = i F 111 = i Λ ext = 4449 Å E =.89 ev w = 1545 Backscatteing Bagg eflection

16 The Dawin Cuve Dawin width in enegy E Γ F F + F F F E= θ= θ Expeimental boadening sin θb E minimal fo θ B 9 hkl =111 : E =.9 ev hkl = : E =.38 ev

17 X-ay Standing Waves The standing wave field Coheent intefeence of incident and eflected wave iω t ik E (,t) = Ee E (,t) = R( θ) Ee whee K = K + iω t ik +ν θ i ( ) with R = E E I( θ,) = E+ E = E ( 1+ R( θ ) + R( θ) cos( ν θ ( ) ) )

18 The standing wave field X-ay Standing Waves Phase shift esults in a shift of nodes in the XSW field

19 istoical milestones - Expeiments XSW expeiments 1964 Batteman Fist XSW expeiment: Ge() single cystal diffaction & fluoescence measuement Ge() cystal 1969 Batteman XSW study of impuity atoms in As in Si(111) 198 Cowan XSW study of an adsobate B/Si(111) 198 Golovchenko XSW tiangulation: B/Si(111) 1984 Babee XSW by peiodic multilayes with lage d spacing 1988 Wooduff Nomal incidence XSW on Cl/Cu(111) LB film on Au 1989 Bedzyk TER-XSW: Langmui-Blodgett films on a gold mio 4 Okasinski XSW atomic imaging by FOURIER invesion: Sn/Ge(111) 5 Gelach XSW on lage oganic molecules F 16 CuPc/Cu(111) Sn/Ge(111)

20 ZnPc/Cu(111) XSW expeiments on ZnPc/Cu(111) Results: small bonding distance d(c) =.41Ǻ Pc coe almost plana d(c) d(n) cental Zn atom below Pc plane d(zn) =.5 Ǻ

21 Futhe Infomation Softwae and databases on X-ay popeties enke tables: f 1, f Come Mann X-ay seve X-ay Data booklet XOP Liteatue: Intoductoy: J. Als-Nielsen & D. McMoow, Elements of Moden X-ay Physics, Wiley, Chicheste 1 (chap. 5) B. E. Waen, X-ay Diffaction. Dove Publications, 199 (chap. 14) Advanced: A. Authie, Dynamical Theoy of X-Ray Diffaction, Oxfod Univesity Pess, Oxfod 1 B. W. Batteman, Dynamical Diffaction of X-ays by Pefect Cystals, Rev. Mod. Phys. 36(3)

22 Applications The Bomann Effect (1941) Anomalous tansmission (fowad diffaction) T µ d I = I e? µ d 1 µd <<1 µ d 1 The Poblem: Popagation of X-ays in pefect cystals?