November 6, 2009: Cheiron School 2009 @ SPring-8

Small-Angle X-ray Scattering
Basics & Applications

Yoshiyuki Amemiya and Yuya Shinohara
Graduate School of Frontier Sciences,
The University of Tokyo

Overview

IntroductionWhat’s SAXS ?

HistoryApplication field of SAXSTheoryStructural Information obtained by SAXS

Experimental Methods
OpticsDetectors

Advanced SAXSMicrobeam, GI-SAXS, USAXS, XPCS etc...
2

What’s Small-Angle X-ray Scattering ?
scatteredx-ray2θ

transmittedsample

area detector
Imaging Plate,CCD etc.

beam stop

Bragg’s law:small angle

large structure(1 - 100 nm)

crystalline sample --> small-angle X-ray diffraction: SAXD
solution scattering / inhomogeneous structure --> SAXS
3

History of SAXS (< 1936)
Krishnamurty (1930)
Hendricks (1932)
Mark (1932)

Observation of scatteringfrom powders, fibers, and colloidal
dispersions

Warren (1936)

carbon black

Molten silica - silica gel
courtesy to Dr. I.L.Torriani

4

History (> 1936)
A. Guinier (1937, 1939, 1943)

Interpretation of inhomogeneities in Al alloys“G-P zones”, introducing the concept of “particlescattering” and formalism necessary to solve
the problem of a diluted system of particles.

O. Kratky (1938, 1942, 1962)
G. Porod (1942, 1960, 1961)
Description of dense systems ofcolloidal particles, micelles, andfibers.
Macromolecules in solution.

Single crystals of AlCu hardened alloy

courtesy to Dr. I.L.Torriani

Hemoglobin
5Application of SAXS

Typical SAXS image
gel

Proteins in solution (Dr. Svergun, EMBL)

100 nmNanocomposite
6Application of SAXSSize and form of particulate system



Inhomogeneous structure

Colloids, Globular proteins, etc...
Polymer chain, two-phase system etc.

Distorted crystalline structure

Crystal of soft matter

7

SAXS of particulate systemsmaller structure

Scattering Intensity I(q)larger structure

Form of particle
I(q) ~ q-1 (rod)~ q-2 (plate)
Surface structureInterparticle

I(q) ~ q-(6 - ds)(ds: surface fractal dimension)

Size of particle

Inter-atomic structure(WAXD)

I(q) ~ exp (-q2Rg2/3)(Rg: radius of gyration)

Scattering angle 2θ or Scattering vector qq = 4π sin θ /λ8

Basic of X-ray scattering
Scattered X-ray(wavenumber: ks)
Incident X-ray

Scattering vectorks(wavenmber: ki)r2θq = ks - kiρ(r) : electron densitykiSampleq = |q|= 4π sin θ /λAmplitude of scattered X-ray
Fourier transform of electron density

Scattering intensity per unit volume:

9

Correlation Function & Scattering Intensity
Correlation function of electron density per unit volume(Debye & Bueche 1949)

Patterson Function

asymptotic behavior of the correlation function

Scattering Intensity : Fourier Transform of correlation function

10

Real space and Reciprocal SpaceReal Space
Fourier Trans.

Scattering amplitude
A(q)

ρ(r)

γ(r)
Autocorrelation
Function

Inv. Fourier Trans.

Squaring

Autocorrelation

Electron Density

Reciprocal SpaceI(q)Scattering Intensity
11Diffraction from Lamellar Structure
ρeL

ideal orderingdcγ (r)Q

ρc - ρadc
L
ρer

γ (r)
QLong period changes.

ρc - ρadcr
ρe

γ (r)
Q

Thickness of crystal changes.

ρc - ρadcr

real space

autocorrelation
12

Normalized Correlation Function
Local electron density fluctuations:average density fluctuaitons
Normalized Correlation Function

substitution

Only the average density fluctuations
contribute to the signal.

Not observable.
13

Invariant Q

Parseval’s equality

Omitted.

Parseval’s equality
Fourier Trans.

Invariant:14

Spherical sample
ρ (r)

I (q)

Fourier Transform& squaredrq

15

Homogeneous sphere

isotropic scattering
courtesy to Dr. I.L.Torriani

16Homogeneous elipsiod
Fixed particle

Random orientation

anisotropic scattering

isotropic scattering

courtesy to Dr. I.L.Torriani17

Size distribution

Scattering Intensity

Larger distribution

01234-1q/Å

0.1 %
1%
3%
5%
7%
10 %15 %20 %30 %5Based on Gaussian distribution

When the form has distribution,
fringes are missed.

6x10-218

Radius of Gyration -- Guinier Plot

Guinier plot: log (I(q)) vs q2

O. Glatter & O. Kratky ed., “Small Angle
X-ray Scattering”, Academic Press (1982).
19

Scattering Intensity I(q)Structure Factor & Form Factor
Form of particle
I(q) ~ q-1 (rod)~ q-2 (plate)
Surface structureInterparticle

I(q) ~ q-(6 - ds)(ds: surface fractal dimension)

Size of particle

Structure Factor Form Factor

Inter-atomic structure(WAXD)I(q) ~ exp (-q2Rg2/3)(Rg: radius of gyration)

Scattering angle 2θ or Scattering vector qq = 4π sin θ /λintra-particlestructureinter-particle structure

Separation of S(q) & F(q)Everlasting issue

Proposed remedy:(especially, for non-crystalline sample)•GIFT (Generalized Inverse Fourier Trans.) by O. Glatter20

Scattering from Inhomogeneous Structure
Electron Density

Autocorrelation Function!

rρ̃(r) = exp −ξ

two phase system
Autocorrelation

Scattering Intensity"

Debye-Bueche1I(q) =(1 + ξ 2 q 2 )2Fourier trans.

!

ξrρ̃(r) = exp −rξ"Ornstein-Zernike

1I(q) =1 + ξ2 q2

polymer chain etc.
21Two-phase system
Phase 1: ρ1, volume fraction ϕ Phase 2: ρ2 volume fraction 1 - ϕBabinet’s principle
Two complementary structures produce the same scattering.

22

Two-phase system -- cont.
Averaged square fluctuation of electron density
whereInvariant: does not depend on the structure of the two phases but
only on the volume fractions and the contrast between the two phases.
23

Porod’s law
For a sharp interface, the scattered intensity decreases as q-4.

internal surface area
Combination of Porod’s law & Invariant

surface-volume ratio
important for the characterization of porous materials

24Intensity for random particle system
Scattering intensity:
Pair distance distribution function :PDDF
the set of distances joining the volume elements within a particle,
including the case of non-uniform density distribution.

Particle’s SHAPE and maximum DIMENSION.
histogram o all intra-particle distances

pairs of volumeelements i-j25

Spherical particle

courtesy to Dr. I.L.Torriani

26

Cylindrical particle

courtesy to Dr. I.L.Torriani

27Flat particle

courtesy to Dr. I.L.Torriani

28

Ellipsoids
ProlateOblatecourtesy to Dr. I.L.Torriani29

Two ellipsoid = dimer

courtesy to Dr. I.L.Torriani

30

Diffraction from Periodic Structure
Diffraction from Unit cell (Crystalline structure factor)

Incident X-rayrj

F(q) =!
jf (q) exp(−iq · r j )f (q) : Atomic Form Factor

!
!
!
!
2!!
!! !!
!!2
Diffraction
F(q)
I(q)
∼G(q)
Intensity:
Maximum ~ N2



FWHM ~ 2π/N!!
!!2 sin2 (πNq · r)Laue function: !G(q)! =sin2 (πq · r)

FWHM --> Size of crystal
31

Laue Function!!
!!2 sin2 (πNq · r)Laue function: !G(q)! =sin2 (πq · r)

N = 100

Large crystal
High diffraction intensity



Narrow FWHM

Soft matter (crystal size: small)
Low diffraction intensityWide FWHM
low S/N

N = 50N = 10

Crystal size --> Intensity & FWHM of diffraction32

Imperfection of crystal (2D)Imperfection of 1st kind

Imperfection of 2nd kind

Thermal fluctuation etc.

in the case of soft matter

33

Imperfection of crystal
Imperfection of 1st kind

Imperfection of 2nd kind

Autocorrelation34

Imperfection of lattice (1D)(a)
Perfect lattice(b)Imperfection of 1st kind(c)Imperfection of 2nd kindEffect of imperfections on diffraction ?

35

Diffraction from lattice-structureA(q)Diffraction

rjqForm of latticez(r) with imperfection ---> calculate Z(q)36Imperfection of 1st kind: distribution function
Fourier trans.

Diffraction with imperfection:

Thermal fluctuation (p(r): Gaussian)ideal lattice

Debye-Waller factor:- decrease diffraction intensity (no effect on FWHM)- background at larger angle diffraction37

Imperfection of 2nd kind4|Z(q)|2/N

321Paracrystal theory

00123456q / (2d!)

Decrease of diffraction intensity andIncrease of FWHM
R. Hosemann, S. N. Bagchi, Direct Analysis of Diﬀraction by Matter,
North-Holland, Amsterdam (1962).

38

X-ray Source for SAXSBrilliance -- Product of size and divergence of beam
beam divergence[photons/(s⋅mrad2⋅mm2⋅0.1% rel.bandwidth)]

Brilliance is preserved (Liouville’s theorem).

beam size

SAXS with a low divergence and small beam
High brilliance beam is required !
39

SAXS Optics
Ge Bent Monochromator

PF BL-15A

Storage-Ring
2.5 GeV
250-400 mA

Bent Mirror

SampleFocal Point

X-ray CCD Detector

PF BL-10C

40

SAXS slits

41

Detectors for SAXSGood Point

Drawback

PSPC•time-resolved•photon-counting•low noise•counting-rate

ImagingPlate•wide dynamic range•large active area

•slow read-out•time-resolved•high sensitivity•image distortion•low dynamic range•fast read-out•automated measurement•not good for time-

CCDwith Image
Intensifier

Fibertapered
CCD

limitation

resolved42

X-ray CCD detector with Image Intensifier
Image
Intensifier

Lens
systemVacuumCooledCCD

Coolingwater
In OutReadout
Unit

e-Analog I/F

DriverControl Unit
&14 bit A/De-Be
CsI:NaRbCsSb

Phosphor screen
CCD(Zn, Cd)S:Ag

Gate High-voltage
power-supply

RS-232C
Thermoelectro
coolerDigital I/FPentium Pro
200MHz256MB MemoryDataacquisition
system

IC-PCI included
Monitor43

Advanced SAXSMicrobeam X-ray

Time-resolved- Inhomogeneity of nano-structure- time evolution of structure- local time evolution of structure

GI-SAXS- surface, interface,thin films

SAXS

XPCS
- structural fluctuation- dynamics

Combined measurement with DSC, viscoelasticity
2D measurement
wide-q (USAXS-SAXS-WAXS)- hierarchical structure- anisotropic structure44

Application of paracrystal theory
Collab. with Kao ltd.

Single Fiber(30 - 100 µm)

African

Caucasian

Asian
X-ray Microbeam( 5 µm x 5 µm)

Relationship between macroscopic form
and microscopic structure?
Local observation with an X-ray microbeam

45

Internal structure of wool

α-helixSEM 像

Coiled-coil dimer

CuticleMatrixIntermediate Filament - 10 nm（ globular protein ）CortexR. D. B. Fraser et al., Proc. Int. Wool Text. Res.
Conf., Tokyo, II, 37, (1985) partially changed.

H. Ito et al., Textile Res. J. 54, 397-402 (1986).

Relationship between IF distribution
and hair curlness?
46

Structure of Intermediate Filament
Scattering pattern

1D intensity profile

Real space structure
DiameterFiber axis

IF<d>IF-IF Distance
Fibre Axis
0˚180˚IF Tilt Angle
FWHM0˚90˚180˚IF47

Diffraction intensity profiles

innerABCD

outer

scattering intensity [a.u.]

Diffraction peak originating from IFInnerABCDOuter
scattering vector s = 2 sinθ/λ (Å-1)

Difference in diffraction intensity--> Structural difference in cortex.
48

Deformation process of spherulite
drawing direction
130

1100400˚
direct beam

POM
(a)

SAXS
(b)

WAXD
(c)
BL40XU @ SPring-8

Local deformation manner of polypropylene during uniaxial elongation process
Combined measurement of polarized microscopeand microbeam SAXS/WAXD.
50

(a)

(b)

(c)

(d)

(e)

Deformation model of PP(b)(a)

(c)

a*(d)

(a)

(b)

(c)

Y. Nozue, Y. Shinohara, Y. Ogawa et al., Macromolecules, 40, 2036 (2007).
52

Grazing Incidence SAXSAdvantage
Surface/interface sensitive (beam footprint).
In-plane structure and out-of-plane structure can be separated.
Thin film sample on substrate can be measured.

Ex: from Web page of Dr. Smiligies @ CHESS

AFM image

GI-SAXS image
53

USAXS using medium-length beamline

sample- 160 mdetector

54

USAXS patterns from elongated rubber
Rubber filled with spherical silica
5TEM image

Stress [MPa]

432

10050

100

strain [%]150200elongation
Scattering pattern also shows hysteresis.
Y. Shinohara et al., J. Appl. Cryst., 40, s397 (2007).

55

Structural information from USAXS

Scattering Intensity

10 5
104

103

102

Size of agglomerate
Mass fractal
dimension of agglomerate dm
I(q) ~ q-dm

largerstructure

Surface fractal dimensionof aggregate/particle ds
I(q) ~ q-(6-ds)

10 1
10 010-1

10-2

Size of aggregate

24

6 80.001

24

6 8-1q /Å0.01

24

6 80.156

X-ray Photon Correlation Spectroscopy: XPCS
Measurement of fluctuation of X-ray scattering intensity--> Structural fluctuation in sample

!"#$%&'()&*+,(-'$,,./&#*((0(!)-(10(2+3,3#('3//.4$,&3#(56.',/35'36"(7((8#,.#5&,"(94:',:$,&3#(56.',/35'36" 14)&*"5(*+66%)/'+"&(speckle71.$/80$9(pattern.)''$%&!"#$%$&'((%)*+)'+"&!

$,-,(0)1$%(0+-#'∗!I(q,0)I(q, τ)#!(2)g (q, τ) =3/)''$%+&-(:$/'"% !
!I(q)#2!"#"$#%$,-,(.)%'+/0$(121.$&1+"&!""""""""""""""

%"

$"#$

>1(1/)''$%+&-(5$*+25($:"0:$1?(Fluctuation of intensity;&'$&1+'<( 1%(!21.$/80$(.)''$%&(602/'2)'$1)2'"/"%%$0)'+"&(62&/'+"&Scattering Intensity()*+,-)*./)*+% ";&'$&1+'<()'(15)00(4*$'$/'"%

1%(2!

%!

) + '#0& (*!

1.030@$0)<('+5$(((((&3=+5$((((((((((((((((((((($50100150Time /s200

relaxation time in system'()*

Autocorrelation

'()*

0!

Time-resolved SAXS with coherent X-ray

3/%$$&1%&!&$')%&!&$,&'2

5%&!&$'!

Autocorrelation4)&*"5(1/)''$%$%

3'%2/'2%$("6(!"#$%&$'!(#$)*+(##$%!",)-."+#!/"?(+&#&&'0 -+:$1(+&6"%5)'+"&("&(1/)''$%$% *<&)5+/1

1.020
1.010
1.000
124

6 8

2

10Time /sec

4

6 8

100

257

Scattering Intensity

Dynamics of nanoparticles observed with XPCS

coherent x-ray

100nm

nano-particles in rubber

5430speckle pattern

50100150
Time /s

200

fluctuation ofscattering intensity

Dependence of dynamics on...
1.030

Autocorrelation- Volume fraction of nano-particles- Vulcanization (cross-linking)- Type of nano-particles- Temperature
etc.

1.020
1.010
1.000
12

46 82

10Time /sec

46 8100

2

Dynamics of Filler in Rubber
58

WAXSt/s

103

10-3

10-6

100

10-1

USAXS
10-2

10-3

10-4

10-5

E /eV
X-ray Photon Correlation
Spectroscopy

10Photon Correlation"cold"

Neutron Scattering
10-910-1210-15q/nm-1NSEBrillouin
Raman

TOF
Inelastic X-ray Scattering
100-15

10-12

10-910-6

BS10-1

Light Scattering

100101SAXS

101102103

104

105

10-3

100r/nmBibliography



A. Guinier and A. Fournet (1955) “Small angle scattering of X-rays” Wiley &Sons, New York. out-of-print

O. Glatter and O. Kratky ed. (1982) “Small Angle X-ray Scattering” Academic
Press, London. out-of-print

L. A. Feigin and D. A. Svergun (1987) “Structure Analysis by Small Angle X-rayand Neutron Scattering” Plenum Press. out-of-print ?

P. Lindner and Th. Zemb ed. (2002) “Neutron, X-ray and Light Scattering: SoftCondensed Matter”, Elsevier.

Proceedings of SAS meeting (2003 & 2006). Published in J. Appl. Cryst.

R-J. Roe (2000) “Methods of X-ray and Neutron Scattering in Polymer Science”,
Oxford University Press.

60