Small-angle X-ray scattering - Hasylab - Desy

The very basics of
Small-angle X-ray
scattering
Ulla Vainio
11.8.2009
DESY summer school lectures

History
• Beginning of 20 th century x-ray diffraction studies of metals show some strange
scattering in the middle of the scattering pattern. It is later named “small angle
scattering” and becomes a separate field.
• A. Guinier, P. Debye, V. Luzatti, G. Porod and others (1940-1960) develop
interpretation for the basic features in SAXS patterns
• Early 1970s first synchrotron SAXS studies on muscle fibers at DESY, end of 1970s
time resolved studies at DORIS
• Mandelbrot invents the term ‘fractal’ in 1975 and writes a book about it in 1982,
shortly after Bale and Schmidt apply fractal concepts to small angle scattering
(1984) (their paper is cited 498 times 15.7.2009)
• Stuhrmann (1983) builds the first anomalous small-angle x-ray scattering (ASAXS)
beamline in the world at DORIS synchrotron for low x-ray energies and studies
biological systems. The idea was copied from anomalous diffraction.
• Haubold et al. (1994) and Simon and Lyon (1994) give examples of ASAXS in book
Resonant anomalous X-ray scattering
• Svergun and his team at DORIS synchrotron develop SAXS software for data
analysis of SAXS from proteins (early 1990s until now). The software becomes
popular also among other SAXS community, because it’s freely available from the
webpage and is very automatized. Glatter develops simultaneously in Austria another
approach to data analysis. In many laboratories people develop their own software.
• SAXS boom: More than 1.2 million articles have been published in this field and 0.8
million of them after year 1999!
Photo source: A. Guinier (1969), Physics today 22, 25.

20
40
60
80
Small is big & big is small
Real space
100 0
20 40 60 80 100
(after you take a Fourier transform)
1
0.8
0.6
0.4
0.2
20
40
60
80
100
Inverse space
20 40 60 80 100
10
8
6
4
2

20
40
60
80
Small is big & big is small
Real space
100 0
0 50 100
(after you take a Fourier transform)
1
0.8
0.6
0.4
0.2
FFT
-0.3
20
40
0
60
80
Inverse space
0.3
-0.3 0 0.3
Pixel q-pixel (1/pixel)
100
0 50 100
Period d = 10 pixels 1 q-pixel is of length 2π/100 ≈ 0.06 1/pixel
Period d = 2π/q = 2π/(10*0.06) = 10 pixels
x 10 6
8
6
4
2
0

Oriented samples (example)
Organoclay players in polypropylene
Packing material for your food in the grocery store
Small-angle X-ray scattering
Transmission electron microscopy
Ristolainen et al. (2005) J. Polymer Sci. B-Polymer Physics 43, 1892.
Studied volume ratio SAXS:TEM about 1000 000 000:1

20
40
60
80
Real space
100 0
20 40 60 80 100
Experiment
1
0.8
0.6
0.4
0.2
20
40
60
80
100
Inverse space
Center is covered by beamstop
⇒Large structures are not seen
because of the beamstop and
because of limited detector resolution.
Largest structure observed
about 100 nm.
q
20 40 60 80 100
10
Largest q restricted
by the detector size
⇒Smallest structure
about 1 nm
8
6
4
2

Example of a SAXS beamline, B1
SAXS detector
100
200
300
400
500
600
0 100 200 300 400
Reference
samples
WAXS
detector
Samples
Monochromatized
X-ray beam from a
bending magnet on
DORIS synchrotron

SAXS beamline 7T-MPW SAXS at BESSY
Monochromatized
X-ray beam from
a wiggler at
BESSY
synchrotron

0
200
400
600
800
Scattering from a sphere
Real space
1000
200 400 600 800 1000
0
Linear scale
R = 30 pixels
1
0.8
0.6
0.4
0.2
0
200
400
600
800
1000
Inverse space
200 400 600 800 1000
Logarithmic scale
14
12
10
8
6
4
2

50
100
150
200
250
300
350
Projected electron density of a sphere in 3D on 2D plane
Real space
400 0
100 200 300 400
50
40
30
20
10
50
100
150
200
250
300
350
400
Inverse space
ϕ
100 200 300 400
20
15
10
5

Intensity (arbitrary units)
10 10
10 5
10 0
Intensity of a sphere
10 -1
q (1/nm)
Sphere (projected)
Circle in 2d
q -4
q -3
Theoretical
10 0
de = 3
de = 2
Many SAXS curves
follow a power law
I(q) ∝ q -α
α is the power law exponent
d e = Euclidian dimension (3 in real life)

Mass fractal
Menger sponge, mass fractal dimension D m = 2.72
(Source: http://commons.wikimedia.org/wiki/File:Menger_sponge_(Level_1-4).jpg)

Surface fractal
Koch snowflake, surface fractal dimension D s = 1.26 (in d e = 2)
(http://commons.wikimedia.org/wiki/File:KochFlake.png)
In nature the fractals are not mathematically self similar.
Avnir et al. (1998) criticize the use of fractal theory to everything in
their paper in Science “Is the geometry of nature fractal?”
In their opinion the power law should extend more than one order of
magnitude, which may correspond just to one step!

Intensity (arbitrary units)
10 12
10 10
10 8
10 6
10 4
10 -1
Fractals, self similar objects
q (1/nm)
Sphere (projected)
Circle in 2d
q -4
q -3
d e = 2!
D m = mass fractal dimension
D s = surface fractal dimension
d e = Euclidian dimension (3 in real life)
10 0
Many SAXS curves follow a power law
I(q) ∝ q -α
Mass fractal:
α = Dm (0 − 3)
Surface fractal:
α = 2d e –D s (6 − D s , when d e = 3)
Shape D m D s
Line 1 0
Platelet 2 1
Sphere 3 2
Porod law
I(q) ∝ q -4

Form factor & structure factor of particles
• Intensity = form factor x structure
factor I(q) = P(q)S(q)
– This equation works if you have
spherical monodisperse particles,
otherwise the factors may be
more mixed
• Structure factor S(q) is related to
the ordering and distance between
particles
• Form factor tells about the shape
and size of the particles
– In a very dilute solution of
particles I(q) = P(q) because S(q)
= 1
P(q)
Form factor
More on form factors of differently shaped particles:
Jan Skov Pedersen: Analysis of small-angle scattering data from colloids and polymer solutions:
modeling and least-squares fitting. Advances in Colloid and Interface Science 70 (1997) 171-210.
q
P(q)S(q)
S(q)
q
Structure factor
q
Total scattering

Far from ideal
Real systems have nasty effects which disturb the perfect SAXS measurement of particles
• “Concentration effect”: The contribution of
the structure factor increases when the particle
concentration increases (order is increased)
•Concentration series is usually needed
and then one can extrapolate to 0concentration
and determine the particle
shape and size
• “Polyelectrolyte effect”: Charged particles
(negative or positive) in solution give always a
structure factor
•Add salt to suppress the interaction
caused by charges (but this can make the
particles willing to aggregate!)
Problem 1: Why measuring very dilute
solutions is not usually possible?
Problem 2: Why you cannot measure forever
long a sample with a synchrotron beam?
Intensity / concentration
Concentration effect
Concentration
increases
Concentration
increases
q
Polyelectrolyte effect

Scattering from particles
Distance distribution function a.k.a. p(r) function
Svergun & Koch, Rep. Prog. Phys. 66 (2003) 1735–1782
Do you find a mistake here?

Scattering from particles
Distance distribution function a.k.a. p(r) function
Svergun & Koch, Rep. Prog. Phys. 66 (2003) 1735–1782

N(R) = number size distribution
R = size of the particle
Polydispersity*
(*) lot’s of different sized particles
Vainio et al. (2006) Latvian journal of physics and technical sciences 4, 14.

How to perform an experiment?

Example: Ontime chemistry at beamline A2
by Vivian Rebbin
• The formation of micelles is observed
with SAXS. With ongoing
condensation of silica, the ordering
process starts.

Courtesy of Vivian Rebbin
On-time chemistry with SAXS
Sample containing the triblock copolymer Pluronic P123 as structuredirecting
agent, 1,4-bis(triethoxysilyl)benzene as silica source, pH = 1,
increasing temperature
Structure: From diffuse scattering to 2d hexagonal

Example: Spinodal decomposition
in amorphous metals
3d atom probe of yttrium atoms
N. Mattern et al. Acta Materialia 57 (2009) 903–908

Anomalous small-angle X-ray scattering (ASAXS)
“What is the unit of intensity?!”

Two-phase approximation
• Only mean electron density difference counts
B
A (vacuum)
Absorption
Anomalous
scattering factors
Intensity (log scale)
Conclusion: f’’
Pink atoms are distributed
homogeneously into one phase
(in small-angle scattering)!
f’
Energy
Absorption edge of pink element
I(E1)
I(E2)
I(E1) - I(E2)
Scattering angle (log scale)
Intensity ∝ |ρB (E) - ρA |
Real space Fourier space
2
Atomic scattering factor
Z + f’(E) + if’’(E)

Three or more phases
• One more phase now ASAXS can help
S bb
S pp
S bp
B
A (vacuum)
Absorption
Anomalous
scattering factors
Energy
f’’
f’
Intensity (log scale)
Absorption edge of pink element
I(E1)
I(E2)
I(E1) - I(E2)
Scattering angle (log scale)
Intensity(E) = Spp (E) + Sbp (E) + Sbb Real space Fourier space
Partial structure factors
Assumption: Atoms in phase B have constant f (no absorption edge at this energy)!

Lignosulfonate, the brown stuff in a tree
Trees
Pulp & paper mills
Paper products,
cellulose
Lignin (in the form of
Kraft lignin or lignosulfonate)

Intensity (relative units)
10 10
10 8
10 6
10 4
Simple model
Fourier transform
10 -2 10 2
10 -1
Lignosulfonate ASAXS
Partial structure factors
q (1/nm)
10 0
Rubidium (Rb) counterions
Rb
Rb-LS
LS
Total
10 1
Intensity (arb. units)
10 2
10 1
10 0
10 -1
10 -2
10 -3
10 -4
Comparison to experiment
model
experiment
10 0
q (1/nm)
I(E1)-I(E2) (model)
I(E1) (model)
I(E1)-I(E2) (Rb-LS)
I(E1) (Rb-LS)
10 1

SAXS beamlines at DORIS III
• SAXS at A2, B1
• USAXS at BW4
• ASAXS at B1
• GISAXS at BW4
• Rheology & SAXS BW1

Books about SAXS
(in the order I like them most)
• A Guinier (1956/1994) X-ray diffraction. In crystals, imperfect crystals, and
amorphous bodies. Chapter 10 Small-angle x-ray scattering.
• Glatter & Kratky (ed.) (1982) Small Angle X-ray Scattering.
(http://physchem.kfunigraz.ac.at/sm/Software.htm)
• Feigin & Svergun (1987) Structure analysis by small-angle X-ray and neutron
scattering.
(http://www.embl-hamburg.de/ExternalInfo/Research/Sax/reprints/feigin_svergun_1987.pdf)
• Guinier and Fournet (1955) Small angle scattering of X-rays.
ASAXS review
• D. Tatchev (2008) Structure analysis of multiphase systems by anomalous smallangle
X-ray scattering. Philosophical Magazine 88, 1751 – 1772.

Programs to analyze SAXS data
But be careful: you get always a result. It doesn’t mean it’s right.
Examples of standalone programs that one can get from the web:
• Dimitri Svergun’s ATSAS package at EMBL
http://www.embl-hamburg.de/ExternalInfo/Research/Sax/software.html
• Joachim Kohlbrecher’s SASfit at PSI http://kur.web.psi.ch/sans1/SANSSoft/sasfit.html
• S. Förster’s Scatter at Uni Hamburg
http://www.chemie.uni-hamburg.de/pc/sfoerster/software.html
• Richard Heenan’s FISH at ISIS
http://www.isis.rl.ac.uk/LargeScale/LOQ/FISH/FISH_intro.htm
• G. Fritz’s Singlebody http://physchem.kfunigraz.ac.at/sm/Software.htm
• More programs (standalone and those requiring Matlab or Igor) at
http://www.small-angle.ac.uk/small-angle/Software.html