Methoden moderner Röntgenphysik II Streuung und Abbildung

Transcription

1 Methoden moderner Röntgenphysik II Streuung und Abbildung Stephan V. Roth DESY

2 Outline > 1.5. : Small-Angle X-ray Scattering (SAXS) > : Applications & A short excursion into Polymeric materials > 1.5. : Grazing incidence SAXS (GISAXS) >.6. : The polymer-metal interface application of GISAXS On the route to organic electronics > 4.6.: In-situ studies of metal layer growth Stephan V. Roth Moderne Methoden der Röntgenphysik II Page

3 T-SAXS vs. GISAXS z y x q z q y q x Beamstop i z y x f q z q y q x Si - Easy measurement - Easy analysis - In-plane information (q y,q z ) - Any possible scattering from substrate - Transparency of substrate - High energy Lee et al., Macromolecules, 38, 8991 (5) - Strong intensity - Easy preparation of samples - Full information (q x,q y,q z ) - Scattering from surface / internal structure - Scattering from reflected AND transmitted beam - Refraction effects (DWBA) - Special setup Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 3

4 Aim > To understand the structure property relation of materials on multiple length scales - q-resolution - Maximum q-value - Beam size - Real pieces & materials - Model systems - Nanotechnology GE-Gas-Turbine-Technology-Selectedfor-Pearl-GTL-Project-in-Qatar Courtesy: R. Gilles (TUM) Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 4

5 Outline II - today > SAXS Introduction > Instrumentation Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter. Eds: P. Lindner, Th. Zemb. North Holland Delta Series, Elsevier, Amsterdam () ISBN: PETRA III > Bulk materials Transmission U/SAXS: Porous materials Ni-base superalloys Droplet drying Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 5

6 Cross-section > Differential cross section Detector I Ω I k f L k i Q d Ω Σ Ω 1 σ Ω V=Sample volume - k f k i sin > Scattering occurs due to density differences Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 6

7 WAXS, SAXS, GISAXS Source: Streumethoden zur Untersuchung kondensierter Materie 1996; ISBN Limit Guinier Porod Bragg SAXS WAXS WAXS: Crystal structure d / resolution [Å] SAXS/GISAXS: density fluctuations, precipitates Log Iq) q x d=1 1 1 =1nm =1µm 1 =1µm qx R=1 Log q > R~particles radius > d~interatomic distance > SAXS: < 5 d sin R 1.54Å Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 7

8 Scattering amplitude > Interference in far field ( r ), V, V P k f q kf I, A( r ) A e ikr ( r i )dv r i k i 4 sin k f k ˆ i e i > Phase difference: > Scattering amplitude: > Intensity: Δ 3 = Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 8

9 Form factor and structure factor: Fourier transform p (r ) Single particle: Fourier transformation 3 Particle distribution function G(r) Electron density distribution 3 Scattering amplitudes of the whole arrangement ] 3 i Scattered Intensity = r i Form factor Structure factor Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 9

10 Two-phase model: Dilute systems > Only form of particle relevant > Matrix M, volume fraction Particles P, volume fraction (1-) Electron density: M,P n M,P *f M,P ASAXS f M,P : atomic form factor ( extension of the electron cloud, resonances) n M,P : number density of atoms > Consider M,P as constant resp. R R R >> R Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 1

11 Two phase Model > Scattering amplitude: 3 = 3 3 Δ 3 3 > = ~ > Porod Invariant Q (Porod, 198): Q 3 4Φ1 ΦΔ Ableiten! Mittelung <..> erklären S.5, S.51 > Only dependent on density contrast Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 11

12 Herleitung Porod-Invariante > Siehe Handzettel und Übung > Q-Berechnung Übung Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 1

13 Two phase Model single particle approximation > Amplitude: Δ 3 > Intensity: = > Closer look at Iq for dilute systems: N P independent scatterers > Incoherent sum of intensities: ~ Δ 1 3 R R Δ n P f P n M f M n M f M = M n P f P = P r V P ~R 3 Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 13

14 Two phase Model single particle approximation > Amplitude: Δ 3 > Intensity: = > Closer look at Iq for dilute systems: N P independent scatterers > Incoherent sum of intensities: ~ Δ 1 3 P( q) sin( qr) qr cos( qr) 3 3 ( qr) - Form factor of a sphere of radius R - Isotropic scattering Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 14

15 Colloid: homogeneous sphere of radius R A simple, but important calculation: F( q) ( r ) e V particlevolume iqr d 3 r R e iqr r sin( ) dddr R R iqr iqr iqr cos( ) e e e r sin( ) dddr r sin( ) qr F( q) dr F( q) F( q) q 4 q R sin( qr) r R dr 4 q qr q r cos( qr) q R R cos( qr) q qr qr 3 dr cos( ) sin( ) 3 sin( ) cos( ) 4R q qr qr qr Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 15

16 Colloid: homogeneous sphere of radius R PqR PqR qr qr Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 16

17 Guinier radius > Q > Homogenous sphere of radius R sin 3 3 ~1 1 5 ~exp 1 5 Ableiten > Radius of gyration: replace homogenous sphere by shell of same moment of intertia: R g > > ~exp 1 general form of Guinier law [Guinier (1955)] 3 > Independent of particle form Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 17

18 Guinier Approximation I(q) qr lim I( q) q V exp( q R g 3 ) Radius of Gyration R g Monodisperse spheres of radius R: R g 3/ 5 R nm Colloids domains Roth et al., Appl. Phys. Lett. 91, (7) Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 18

19 Porod s law: large q Scattered intensity: ~ 4R 3 sin( qr) qr cos( qr) 3 qr Look at maxima of form factor ~ 4 sin( qr) qrcos( qr) qr 3 IQR 4 sin( qr) qr cos( qr) 3 qr ~ ~ 1 qr 4 ~ 4 1 R S 4 ~ q 4 6 R VP q qr 3 qr qr 3 QR Surface of sphere Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 19

20 Porod s Law IqR log I [a.u.] SAXS q^-4 q^-4 USAX qr log q [nm-1] qr R>1µm R~18nm 4.5 > Depends only on Surface and particle Volume > No shape dependance Stephan V. Roth Moderne Methoden der Röntgenphysik II Page

21 The structure factor many particles, close distance > Real systems: not dilute, many particles > Generalisation of Bragg s Law in crystallography: Iq c Pq Sq Form factor Structure factor R Interference due to assembly of particles D max, > Periodic ordering with periodicity d, in the electron density : > Iq shows a corresponding maximum at q/(d max, ) Smearing Lode (1998) Roth et al., J. Appl. Cryst. 36, 684 (3) Distance of particles Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 1

22 The structure factor many particles, close distance > Real systems: not dilute, many particles > Generalisation of Bragg s Law in crystallography: Iq c Pq Sq > Examples: R=5nm, D max =1nm, 5nm, D/D max =5% Iq Low Pq, Sq1 High SqPq q nm 1 q nm 1 Stephan V. Roth Moderne Methoden der Röntgenphysik II Page

23 Structure factor and form factor > D max =5nm D max =1nm > D = 5nm, 1nm,.1nm > 1 well separated particles Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 3

24 Colloidal systems > Latex spheres in water Iq c Pq Sq Low High Pq, Sq1 SqPq > Gaussian distribution of particle sizes > Shift in maximum: Decreasing distance q [nm -1 ] Hu et al., Macromolecules, 41, 573 (8) Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 4