31.05. : Small-Angle X-ray Scattering (SAXS) 02.06. :...
Transcript of 31.05. : Small-Angle X-ray Scattering (SAXS) 02.06. :...
Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
> 31.05. : Small-Angle X-ray Scattering (SAXS)
> 02.06. : Applications &
A short excursion into Polymeric materials
> 04.06. : Grazing incidence SAXS (GISAXS)
2 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
T-SAXS vs GISAXS
- Easy measurement
- Easy analysis
- In-plane information (qy,qz)
- Any possible scattering from
substrate
- Transparency of substrate
- High energy
Lee et al., Macromolecules, 38, 8991 (2005)
- 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
qy
qx
qz z y
x
2q
Beamstop
af
2q x
y z
ai
qy
qx
qz
3 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
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
http://news.thomasnet.com/companystory/
GE-Gas-Turbine-Technology-Selected-
for-Pearl-GTL-Project-in-Qatar-495497
Courtesy: R. Gilles (TUM)
4 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
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
5 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
Cross Section
> Differential cross section
> Scattering occurs due to density differences
𝑑𝜎 =𝐼
𝐼0𝐿2𝑑Ω
𝑑𝜎
𝑑Ω=𝐼
𝐼0𝐿2 ⇨
𝑑Σ
𝑑Ω=1
𝑉
𝑑σ
𝑑Ω
V = Sample volume
𝑞 = 𝑘𝑓-𝑘𝑖
2 if kk
𝑞 = 2
2𝜋
𝜆sin(𝜃)
dW
Q 2q
kf
ki
Detector → I
I0
2
L
6 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
WAX, SAX, GISAX
Source: Streumethoden zur Untersuchung kondensierter Materie
1996; ISBN 978-3-89336-180-9
SAXS WAXS
Limit Guinier Porod Bragg
Lo
g I
(q)
WAXS:
Crystal
structure
SAXS/GISAXS:
density fluctuations, precipitates
d / resolution [Å]
1 10
=1nm
100 1000 10000
=1µm 100000
=10µm
Log q 𝜆 = 2𝑑 sin 𝜃
𝜆 = 1.54Å > R~particles “radius“
> d~interatomic distance
> SAXS: q < 5°
d
R
7 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
Scattering Amplitude
> Interference in far field
> Phase difference:
> Scattering amplitude:
> Intensity:
Δ𝜑𝑖 = 𝑘𝑓 − 𝑘𝑖 ∙ 𝑟 𝑖 = 𝑞 ∙ 𝑟 𝑖
𝐴 𝑞 = 𝜌 𝑟 𝑒−𝑖𝑞𝑟 𝑑𝑉 = 𝜌 𝑟 𝑒−𝑖𝑞𝑟 𝑑3𝑟
𝐼 𝑞 = 1
𝑉𝐴 𝑞 2
q kf
q
q
ki
kf
2q
8 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
Form factor and structure factor: Fourier transform
Single particle: Fourier transformation
𝐴 𝑞 = 𝜌𝑃 𝑟 𝑒−𝑖𝑞𝑟 𝑑𝑉 = 𝜌𝑃 𝑟 𝑒
−𝑖𝑞𝑟 𝑑3𝑟
Particle distribution function G(r) Electron density distribution
𝜌 𝑟 = 𝜌𝑃 𝑟 𝑖𝑖
= 𝜌𝑃 𝑟 ′ 𝐺(𝑟 − 𝑟 ′)𝑑3𝑟′ = 𝜌𝑃 𝑟 ∗ 𝐺(𝑟 )
Scattering amplitudes of the whole arrangement
𝐴 𝑞 = 𝜌 𝑟 𝑒−𝑖𝑞𝑟 𝑑𝑉 = [𝜌𝑃 𝑟 ∗ 𝐺(𝑟 ) ]𝑒−𝑖𝑞𝑟 𝑑3𝑟
= 𝜌𝑃 𝑟 𝑒−𝑖𝑞𝑟 𝑑𝑉 ∙ 𝐺 𝑟 𝑒−𝑖𝑞𝑟 𝑑𝑉
Scattered Intensity
𝐼 𝑞 = 1
𝑉𝐴 𝑞 2 = 𝑃 𝑞 𝑆(𝑞 )
Form factor Structure factor
)(rp
i
ri
9 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
Two phase Model: Dilute systems
> Only form of particle relevant
> Matrix M, volume fraction F
Particles P , volume fraction (1-F)
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
ASAX
2R
2R
x >> R
10 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
Two phase Model
> Scattering amplitude:
𝐴 𝑞 = 𝜌 𝑟 𝑒−𝑖𝑞𝑟 𝑑3𝑟 = 𝜌𝑀 𝑟 𝑒−𝑖𝑞𝑟 𝑑3𝑟
Φ𝑉+ 𝜌𝑃 𝑟 𝑒
−𝑖𝑞𝑟 𝑑3𝑟 (1−Φ)𝑉
𝐴 𝑞 = 𝜌𝑀 − 𝜌𝑃 𝑒−𝑖𝑞𝑟 𝑑3𝑟
Φ𝑉
𝐴 𝑞 = Δ𝜌 𝑒−𝑖𝑞𝑟 𝑑3𝑟 Φ𝑉
> 𝐼 𝑞 = 1
𝑉𝐴 𝑞 2~∆𝜌2
> Porod Invariant Q (Porod, 1982):
Q = 𝐼 𝑞 𝑑3𝑞 = 4𝜋Φ(1 − Φ)Δ𝜌2
> Only dependent on density contrast D
Ableiten!
Mittelung <..> erklären S.25, S.51
11 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
Herleitung Porod-Invariante
> Siehe Handzettel und Übung
> Q-Berechnung Übung
12 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
Two phase Model – single particle approximation
> Amplitude: 𝐴 𝑞 = Δ𝜌 𝑒−𝑖𝑞𝑟 𝑑3𝑟 Φ𝑉
> Intensity: 𝐼 𝑞 = 1
𝑉𝐴 𝑞 2
> Closer look at I(q) for dilute systems: NP independent scatterers
> Incoherent sum of intensities:
𝐼 𝑞 ~𝑁𝑃𝑉𝑃2Δ𝜌2
1
𝑉𝑃 𝑒−𝑖𝑞𝑟
𝑉𝑃
𝑑3𝑟
2
2R 2R
nPfP
nMfM Δ𝜌
r
VP~R3
nMfM
=M
nPfP=
P
13 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
Two phase Model – single particle approximation
> Amplitude: 𝐴 𝑞 = Δ𝜌 𝑒−𝑖𝑞𝑟 𝑑3𝑟 Φ𝑉
> Intensity: 𝐼 𝑞 = 1
𝑉𝐴 𝑞 2
> Closer look at I(q) for dilute systems: NP independent scatterers
> Incoherent sum of intensities:
𝐼𝑚 𝑞 ~𝑁𝑃𝑉𝑃2Δ𝜌2
1
𝑉𝑃 𝑒−𝑖𝑞𝑟
𝑉𝑃
𝑑3𝑟
2
2
3)(
)cos()sin(3)(
qR
qRqRqRqP
- Form factor of a sphere of radius R - Isotropic scattering
14 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
Colloid: homogeneous sphere of radius R
A simple, but important calculation:
drddrerderqF rqi
R
lumeparticleVoV
rqi qq
)sin()()( 2
0
2
0 0
0
3
R iqriqriqr
R
drrqr
eedrddreqF
0
2
0
2)cos(
0 0
0 )sin(2)sin(2)( qqq q
RRR
drq
qr
q
qrr
qdrrqr
qqF
00
0
0
0
)cos()cos(4)sin(
22)(
30
3
2
0 )cos()sin(4
)sin()cos(4)(
qR
qRqRqRR
q
qR
q
qRR
qqF
15 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
Colloid:homogeneous sphere of radius R
16 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
Guinier radius
> Q → 0
> Homogenous sphere of radius R
> Radius of gyration:
replace homogenous sphere by shell of same moment of intertia: Rg
> 𝑅𝑔 =35 𝑅
> general form of Guinier law [Guinier (1955)]
> Independent of particle form
𝑃 𝑞 ~exp(−1
3𝑞2𝑅𝑔
2)
𝑃 𝑞 = 3sin 𝑞𝑅 − 𝑞𝑅𝑐𝑜𝑠(𝑞𝑅)
𝑞𝑅 3
2
~1 −1
5𝑞2𝑅2~exp(−
1
5𝑞2𝑅2) Ableiten
17 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
Guinier Approximation
)3
exp()(lim
2
222
0
g
q
RqVqI D
Radius of Gyration Rg
Monodisperse spheres of radius R: RRg 5/3Roth et al., Appl. Phys. Lett. 91, 091915 (2007)
2nm Colloids
domains
18 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
Porod´s law: large q
Scattered intensity:
2
30
3 )cos()sin(4~
qR
qRqRqRR
I(QR)
QR
Look at maxima of form factor
4
26
2
4
2
30
2
30
2
30
2
30
~1
~
4~1
4~
)cos()sin(4
)cos()sin(4~
qV
S
R
R
q
qR
qR
qR
qR
qR
qRqRqR
qR
qRqRqR
P
Surface of sphere
19 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
Porod´s Law
-1
0
1
2
3
4
5
6
-2.5 -2 -1.5 -1 -0.5 0
log q [nm-1]
log
I [
a.u
.]
SAXS
q -̂4
q -̂4
USAX
I(qR)
qR
R>1µm R~18nm
𝑃 𝑞𝑅 > 4.5 = 2𝜋𝑆
𝑉𝑃2𝑞−4
> Depends only on Surface and particle Volume
> No shape dependance
20 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, R. Roth
The structure factor – many particle, close distance
> Real systems: not dilute, many particles…
> Generalisation of Bragg‘s Law in crystallography:
I(q)= c P(q) S(q)
> Periodic ordering with periodicity d,x in the electron density :
> I(q) shows a corresponding maximum at q=2/(Dmax , x)
𝑆 𝑞 ∝ 1 − exp(−2𝜎 𝑞²𝐷
2 )
1 − 2 exp −𝜎 𝑞²𝐷2 cos 𝑞𝐷𝑚𝑎𝑥 + exp(−2𝜎𝐷
2𝑞2)
Form factor Structure factor
Interference due to assembly of particles
Lode (1998)
Roth et al., J. Appl. Cryst. 36, 684 (2003)
2R
Dmax,x
Smearing Distance of particles
21 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, R. Roth
The Structure factor – many particle, close distance
> Real systems: not dilute, many particles…
> Generalisation of Bragg‘s Law in crystallography:
I(q)= c P(q) S(q)
> Examples: R=5nm, Dmax=100nm, 25nm, sD/Dmax=25%
Low F P(q), S(q)=1 High F S(q)P(q)
q [nm-1] q [nm-1]
22 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
Structure factor and form factor
> Dmax=25nm
Dmax=10nm
> sD= 5nm, 1nm, 0.1nm
> 𝑆 𝑞 1𝑞 ∞
well separated particles
23 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,
SoSe 2016, S. Roth
Colloidal System
> Latex spheres in water
I(q)= c P(q) S(q)
Low F P(q), S(q)=1 High F S(q)P(q)
q [nm-1]
> Gaussian distribution
of particle sizes
> Shift in maximum:
Decreasing distance
Hu et al., Macromolecules, 41, 5073 (2008)