Methoden moderner Röntgenphysik II Streuung und Abbildung

Transcription

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

2 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

3 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 3

4 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 4

5 Colloid: homogeneous sphere of radius R PqR PqR qr qr Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 5

6 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 6

7 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 7

8 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 8

9 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 9

10 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

11 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 11

12 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 1

13 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 Hu et al., Macromolecules, 41, 573 (8) q [nm -1 ] Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 13

14 Illustration > USAXS at photonic crystals > USAXS in highly concentrated colloidal suspensions Beamstop 7 nm onic_table.html Courtesy: V. Boyko (BASF) Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 14

15 Outline > SAXS Introduction > Instrumentation PETRA III > Bulk materials Transmission U/SAXS: Porous materials Ni-base superalloys Droplet drying Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 15

16 SAXS collimation and scattering geometry Detector Collimating Slit Collimating Slit 1 bs min L SD Sample L SD determines resolution min =bs / ( L SD ) Use Bragg s law: d max = min Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 16

17 Layout Different µfocussing schemes > Flexible choice of beam size and divergence > Fixed focal spot position and size E=13keV x1µm E=15keV 4x17µm > Full user operation within design values! CRL System 1 CRL System ~5x1 11 Ph/sec 76.7m 85m 81.7m 86.4m Det 4xµm > Nanofocus end station: 3x3µm <1.5x1.5µm 5x5nm Krywka, SVR et al., J. of Appl. Cryst. 45, 85 (1) x1µm 8x8µm <xµm Roth et al., J. Phys.: Cond. Matter 3, 548 (11) Buffet, SVR et al., J. Synchr. Rad., 19, 647 (1) Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 17

18 Rapid Change (GI)SAXS / (GI)WAXS 1 > Adjust scattering angles dω q-ranges > 5cm<D SD <8.6m > Highly flexible > Separate WAXS device Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 18

19 µusaxs focus > Beam size: 3x3µm > SDD=847mm > N =1 > PS particles: 4nm Dried on glass slide t acq =1s background corrected Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 19

20 Outline > SAXS Introduction > Instrumentation PETRA III > Bulk materials Transmission U/SAXS: Porous materials Ni-base superalloys Droplet drying Stephan V. Roth Moderne Methoden der Röntgenphysik II Page

21 Ni-base superalloys Ni-base superalloys > Ni-base W-rich experimental single crystal superalloy (Ni-4.6Al-6.4Ta-5.7Cr-1.8W-.1Mo) > Ni-Al solid solution Matrix (), fcc > Precipitates ( Al, ), Ni 3 (Al,Ti) > TEM: -precipitates R > 5 nm > D > 1 nm Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 1

22 ID.75 nm -1 Fit noc_4: P. Strunz et al., J. Appl. Cryst. 36, 854 (3) Courtesy: Gilles Strunz Only phase visible : cuboids, round edges D max =(139± 59) nm R = 5 nm (consistent with TEM) Intensity [a.u.] 1 1,E+5 1 1,E+4 1 1,E+3 1 1,E+ 1 Beam: 1m resolution form factor Structure factor Maximum Stephan V. Roth 1,E+1 Moderne Methoden der Röntgenphysik II Page q [nm -1 ] ,E- 1,E-1 1,E+

23 Local precipate morphology µsaxs [3] [3] [1] R. Gilles et al. Scripta Mat. 39, 715 (1998) > phase precipitate: embrittlement of alloy crack formation and propagation - streaking: correct orientation - phase: stack - distance 5-15 m diameter R < 1 m thickness t > O(1 nm) y = 5 m k [1] Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 3-1 z = 5 m Q=1.5 nm

24 Microfocus: local - particle size distribution Roth et al., Nucl. Instr. Meth. B, 55 (3) Porod-law Q -4 SAXS: TEM: USAXS: R > 55 nm R 5 nm R = 5 nm I [a.u.] 1 1 resolution Q [nm -1 ] Lower minimum of particle size distribution Stephan V. Roth Moderne Methoden der Röntgenphysik II Page 4