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Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 2
Outline
> 12.05. : Small-Angle X-ray Scattering (SAXS)
> 19.05. : Applications & A short excursion into Polymeric materials
> 21.05. : Grazing incidence SAXS (GISAXS)
> 02.06. : The polymer-metal interface – application of GISAXSOn the route to organic electronics
> 04.06.: In-situ studies of metal layer growth
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 3
T-SAXS vs. GISAXS
- Easy measurement- Easy analysis- In-plane information (qy,qz)- Any possible scattering from
substrate- Transparency of substrate- High energy
- Strong intensity- Easy preparation of samples- Full information (qx,qy,qz)- Scattering from surface / internal
structure- Scattering from reflected AND
transmitted beam- Refraction effects (DWBA)- Special setup
Si
x
yz
qx
qyqz
2
Beamstop x
y
z
qx
qyqz
2i
f
2
Lee et al., Macromolecules, 38, 8991 (2005)
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 4
> To understand the structure – property relation of materials onmultiple length scales
- Real pieces& materials
- Model systems- Nanotechnology
- q-resolution- Maximum q-value- Beam size
Aim
http://news.thomasnet.com/companystory/GE-Gas-Turbine-Technology-Selected-for-Pearl-GTL-Project-in-Qatar-495497
Courtesy: R. Gilles (TUM)
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 5
Outline II - today
> SAXS – Introduction
> Instrumentation P03/MiNaXS @ PETRA III
> Bulk materials Transmission U/SAXS: Porous materials
Ni-base superalloys
Droplet drying
Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter.Eds: P. Lindner, Th. Zemb. North Holland Delta Series, Elsevier, Amsterdam (2002)ISBN: 0-444-51122-9
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 6
> Differential cross section
> Scattering occurs due to density differences
d
L
2I0
Detector → I
V=Sample volumeQ
ki
kf
2
if kk
Cross-section
2
Ω
Ω ⇨ΣΩ
1 σΩ
- 22
sin
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 7
d / resolution [Å]
1 10=1nm
100 1000 10000=1µm
100000=10µm
WAXS: Crystal
structure
WAXS, SAXS, GISAXS…Source: Streumethoden zur Untersuchung kondensierter Materie1996; ISBN 978-3-89336-180-9
WAXS
qxd=1
2 sin
1.54Å
SAXS SAXS/GISAXS:density fluctuations, precipitates
> R~particles “radius“
> d~interatomic distance
> SAXS: < 5°
Limit Guinier Porod Bragg
Log qd
R
q xR=1
Log Iq)
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 8
Scattering amplitude
( r ), V, VP
0
ir
rkieArA
0)(
ii ek ˆ2
fk
2dVri )(I0,
q
ki
kf kf 2
> Interference in far field
> Phase difference:
> Scattering amplitude:
> Intensity:
Δ ∙ ∙
3
= 2
4sin
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 9
)(rp
Single particle: Fourier transformation
Particle distribution function G(r)
i
ri
Electron density distribution
Scattering amplitudes of the whole arrangement
Scattered Intensity
Form factor Structure factor
Form factor and structure factor: Fourier transform
= 2
3
′ ′ 3 ′ ∗
∗ ] 3
∙
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 10
Two-phase model: Dilute systems
> Only form of particle relevant
> Matrix M, volume fraction Particles P , volume fraction (1-)Electron density: M,P nM,P*fM,P
fM,P : atomic form factor („extension of the electron cloud“, resonances)nM,P : number density of atoms
> Consider M,P as constant resp.
2R
>> R
2R
2R
ASAXS
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 11
Two phase Model
> Scattering amplitude:3 = 3 3
3
Δ 3
> = 2~∆
> Porod Invariant Q (Porod, 1982):Q 3 4 Φ 1 Φ Δ 2
> Only dependent on density contrast
Ableiten!Mittelung <..> erklären S.25, S.51
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 12
Herleitung Porod-Invariante
> Siehe Handzettel und Übung
> Q-Berechnung Übung
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 13
Two phase Model – single particle approximation
> Amplitude: Δ 3
> Intensity: = 2
> Closer look at I q for dilute systems: NP independent scatterers
> Incoherent sum of intensities:
~
Δ 21
3
nPfP=P
nMfM =M
r
2RnPfP
nMfM2R
Δ
VP~R3
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 14
Two phase Model – single particle approximation
> Amplitude: Δ 3
> Intensity: = 2
> Closer look at I q for dilute systems: NP independent scatterers
> Incoherent sum of intensities:
2
3)()cos()sin(3)(
qRqRqRqRqP
- Form factor of a sphere of radius R- Isotropic scattering
~
Δ 21
3
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 15
Colloid: homogeneous sphere of radius R
drddrerderqF rqiR
lumeparticleVoV
rqi
)sin()()( 2
0
2
0 00
3
R iqriqriqr
R
drrqr
eedrddreqF0
20
2)cos(
0 00 )sin(2)sin(2)(
RRR
drqqr
qqrr
qdrrqr
qqF
00
0
00
)cos()cos(4)sin(22)(
A simple, but important calculation:
30
32
0 )cos()sin(4)sin()cos(4)(qR
qRqRqRRqqR
qqRR
qqF
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 16
Colloid: homogeneous sphere of radius RPqR
qR
qR
PqR
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 17
Guinier radius
> Q→ 0
> Homogenous sphere of radius R
> Radius of gyration: replace homogenous sphere by shell of same moment of intertia: Rg
> ⁄
> general form of Guinier law [Guinier (1955)]
> Independent of particle form
3sin
3 ~115
2 2~exp15
2 2
~exp13
2 2
Ableiten
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 18
)3
exp()(lim2
222
0
g
q
RqVqI
Radius of Gyration Rg
Monodisperse spheres of radius R: RRg 5/3
qR
I(q)
2nm Colloidsdomains
Guinier Approximation
Roth et al., Appl. Phys. Lett. 91, 091915 (2007)
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 19
Porod‘s law: large q
Scattered intensity:
2
303 )cos()sin(4~
qRqRqRqRR Look at maxima of form factor
4
26
2
4
2
30
2
30
2
30
2
30
~1~
4~14~
)cos()sin(4
)cos()sin(4~
qVS
RR
q
qRqR
qRqR
qRqRqRqR
qRqRqRqR
P
Surface of sphere
I QR
QR
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 20
> Depends only on Surface and particle Volume
> No shape dependance
qR
I qR
-10123456
-2.5 -2 -1.5 -1 -0.5 0
log q [nm-1]
log
I [a.
u.]
SAXSq -̂4q -̂4USAX
R>1µm R~18nm
Porod‘s Law
4.5 2 2
qR
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 21
> Real systems: not dilute, many particles…
> Generalisation of Bragg‘s Law in crystallography:I q cP q S q
> Periodic ordering with periodicity d, in the electron density :
> I q shows a corresponding maximum at q 2/(Dmax,)
Form factor Structure factor
Interference due to assembly of particles Dmax,
2R
The structure factor – many particles, close distance
Distance of particlesSmearing
Lode (1998)Roth et al., J. Appl. Cryst. 36, 684 (2003)
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 22
> Real systems: not dilute, many particles…
> Generalisation of Bragg‘s Law in crystallography:I q cP q S q
> Examples: R=5nm, Dmax=100nm, 25nm, D/Dmax=25%
The structure factor – many particles, close distance
Low P q ,S q 1 High S q P q
q nm‐1 q nm‐1
Iq
Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 | Page 23
Structure factor and form factor
> Dmax=25nmDmax=10nm
> D= 5nm, 1nm, 0.1nm
> →1 →∞well separated particles
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