September 28, 2012: Cheiron School 2012 @ SPring-8

Graduate School of Frontier Sciences,
The University of Tokyo

Introduction
What’s SAXS ?

HistoryTheory
Basic of X-ray scatteringStructural Information obtained by SAXS

Experimental Methods
X-rray Optics



X-ray Detectors

Advanced SAXSMicrobeam, GI-SAXS, USAXS, XPCS etc...
2Single Molecule（多体中の）単分子t/s

sampleImaging Plate,
CCD etc.

beam stop

Bragg’s law:small angle

large structure

E/eV 10Macrophase separation
マクロ相分離3タ2θ

transmittedarea detector
レオメscatteredx-rayMolecule
Assembly
Organic
分子集合体
相分離構造
vs Structure
組織構造10-1510-1210-910-610-310-0100Micro phaseseparation
ミクロ相分離10-310-610-910-1210-15

合成高分子
Entanglement
絡み合い効果Main chain
motion
主鎖運動

細胞
CellMembrane
膜

Sidechain
側鎖運動10crystalline sample --> small-angle X-ray diffraction: SAXD

生物
Living organism

Segment
motionセグメント運動

Biologicalmacromolecules
生体高分子

SAXS

WAXS-210-110USAXS NFS

ものと生き物のクロスオーバー

0101102103104r/nm
105106107-1s/nm

solution scattering / inhomogeneous structure --> SAXS

10210110010-110-210-310-410-510-610-7

電子顕微鏡

34Krishnamurty (1930)
Hendricks (1932)

Incident X-ray

Mark (1932)

Observation of scatteringfrom powders, fibers, and colloidal
dispersions

Warren (1936)
Each electron in marerials vibrates and emits secondary spherical wave

When there are two electrons,interference between secondary waves from electrons
when a distance between electrons is small--> scattering at large angle is intensiﬁed
when a distance between electrons is large

carbon black--> scattering at small angle is intensiﬁed --> SAXS

Molten silica - silica gel(above )5(below)

courtesy to Dr. I.L.Torriani

6

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.
Typical SAXS image

O. Kratky (1938, 1942, 1962)
gel

G. Porod (1942, 1960, 1961)

Proteins in solution (Dr. Svergun, EMBL)

Description of dense systems ofcolloidal particles, micelles, andfibers.
Macromolecules in solution.

100 nmSingle crystals of AlCu hardened alloy

courtesy to Dr. I.L.Torriani

Hemoglobin
7

Nanocomposite
8

Scattered X-ray(wavenumber:wave vector : ks)

Introduction






Incident X-rayr

History

Theoryp = k

Basic of X-ray scattering

momentum of a photon

Structural Information obtained by SAXS

2θIntensity of scattered X-ray :X-ray Detectors(Extensive variable)

Advanced SAXSIntensity of scattered X-ray per volume:

Microbeam, GI-SAXS, USAXS, XPCS etc...

(Intensive variable)

Importance of phase,Real SpaceReciprocal Space
Fourier Trans.

This doesn’t depend on sample volume, andis used to obtain the absolute intensity

9of complex amplitude
IFT

Scattering amplitude

ρ(r)

A(q)
Inv. Fourier Trans.

Squaring

Autocorrelation

aroundFourier transform of electron density

IFT

Fourier Trans.

γ(r)Autocorrelation
Function= Patterson Function

Complex number

q = |q|= 4π sin θ /λX-ray Optics

Electron Density

ρ(r) : electron density

kiSampleq = ks - ki

Amplitude of scattered X-ray

Experimental MethodsScattering vector

kswave vector : ki)(wavenmber:What’s SAXS ?

I(q)Scattering Intensity
Wiener-Khinchintheorem

1110

Importance of phase,of complex amplitude
Autocorrelation function of electron density

IFT??

Patterson Function

asymptotic behavior of the autocorrelation function

γ(0) = ρ2 γ(∞) = ρ2IFT??

Scattering Intensity : Fourier Transform of autocorrelation function

cf. Wiener-Khinchin theorem

ρeL

ideal orderingdcγ (r)Introducing the relative electron density,Qρc - ρaη(r) , asdc
Lρer

Introducing the normalized autocorrelation function, γ0 (r) , as
1 1
γ0 (r) = 2η(r )η(r + r )drη  V V

γ (r)QLong period changes.

14ρc - ρadcr
ρethen,γ (r)γ0 (r) =γ(r) − ρ2
η2 

where

γ0 (0) = 1 γ0 (∞) = 0

Q

Thickness of crystal changes.

ρc - ρadcthen,rreal space

autocorrelation
15

The average of the relative electron density fluctuations determines themagnitude of I(q). The normalized autocorrelation function 0(r)determines the shape of I(q).

Not observable.
16Invariant:

Introduction(when isotropic)
It doesn’t depend on the structure）What’s SAXS ?

HistoryTheory
Basic of X-ray scatteringStructural Information obtained by SAXS

Experimental Methods

Parsevall’s thheorem

X-ray Optics



X-ray Detectors

Fourier Trans.

Omitted.

Advanced SAXSMicrobeam, GI-SAXS, USAXS, XPCS etc...

17

18

Relation between SAXS pattern and Structural information
smaller structure

larger structure

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

1. Size and form of particulate systemScattering Intensity I(q)（

Colloids, Globular proteins, etc...

2. Correlation length of inhomogeneous structurePolymer chain, two-phase system etc.

3. Lattice parameters of distorted crystals (para-crystal)
4. Degree of crystallinity, crystal size, crystal distortion

Crystalline polymer

19

Surface structureI(q) ~ q-(6 - ds)

Interparticle(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 q
q = 4π sin θ /λ20

2ρ (r)I (q)Fourrier Tra
Fourannsform& squarreddr
0Rq2

isotropic scattering
courtesy to Dr. I.L.Torriani

21Larger distribution

Random orientation
Scattering Intensity

Fixed particle

anisotropic scattering

0.1 %
1%
3%
5%
7%
10 %
15 %
20 %
30 %

22

Based on Gaussian distribution

When the form has distribution,
fringes are missed.

isotropic scattering
0courtesy to Dr. I.L.Torriani

23

12

34-1

q/Å

-2

5 6x1024

r2 ρ(r)drρ(r)drScattering Intensity I(q)Rg

R2g = Guinier plot: log (I(q)) vs q2

Form of particle
II(q)I(q) ~ q-1 (rod)~ q-2 (plate)
Surface structureI(q)II(q) ~ q-(6 - ds)

Interparticle

Structure Factor Form Factor(dds: surface fractal dimension)

Size of particleeInter-atomic structure(WAXD)

II(q)I(q) ~ exp (-q2Rg2/3)

intra-particle
structureinter-particle structure((Rg: radius of gyration))

Scattering angle 2θ or Scattering vector q
q = 4π sin θ /λSeparation of S(q) & F(q)Everlasting issue

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

(especially, for non-crystalline sample)•GIFT (Generalized Inverse Fourier Trans.) by O. Glatter

25

Electron Density

Autocorrelation Function

Scattering Intensity26

1, volume fraction  Phase 2: 2 volume fraction 1 - 

Debye-BuecheI(q) =

two phase system
Autocorrelation

1(1 + ξ 2 q 2 )2Fourier trans.
Ornstein-Zernike

I(q) =

1
1 + ξ2 q2

Babinet’s principle
polymer chain etc.

Two complementary structures produce the same scattering.
2728

Averaged square fluctuation of electron density

For a sharp interface, the scattered intensity decreases asq-4.

where : volume fraction

internal surface area
Combination of Porod’s law & Invariant Q

surface-volume ratio
important for the characterization of porous materials
Invariant: does not depend on the structure of the two phases but
only on the volume fractions and the contrast between the two phases.
2930

Scattering intensity:(when isotropic)

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 of all intra-particle distances

pairs of volume
elements i-j

31

courtesy to Dr. I.L.Torriani

32courtesy to Dr. I.L.Torriani

33

courtesy to Dr. I.L.Torriani

34

courtesy to Dr. I.L.Torriani

35

courtesy to Dr. I.L.Torriani

36

ProlateOblateDiffraction from Unit cell (Crystalline structure factor)

Incident X-ray

F(q) =

f (q) : Atomic Form Factor

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

FWHM ~ 2/N2 sin2 (πNqq · r)Laue function: G(q) =2sin (πq · r)

f (q) exp(−iq · r j )jrj 2

 F(q)


2 sin2 (πNq · r)Laue function: G(q) =sin2 (πq · r)Large crystal
High diffraction intensity



Narrow FWHM

Soft matter (crystal size: small)

Low diffraction intensityWide FWHM
low S/N

FWHM --> Size of crystal

N = 100

N = 50N = 10

Crystal size --> Intensity & FWHM of diffraction
3738

Imperfection of 1st kind

Imperfection of 2nd kindAutocorrelation

Imperfection of 1st kind

Imperfection of 2nd kind

Thermal fluctuation etc.

in the case of soft matter

394040A(q)(a)
Perfect lattice

Diffractionrj(b)Imperfection of 1st kindq(c)Imperfection of 2nd kind

Form of lattice
convolution

Effect of imperfections on diffraction ?

z(r) with imperfection ---> calculate Z(q)

41424



3|Z(q)|2/N





21



00123

456q / (2dDecrease of diffraction intensity andIncrease of FWHM
R. Hosemann, S. N. Bagchi, Direct Analysis of Diractionby Matter, North-Holland, Amsterdam (1962).

43IntroductionWhat’s SAXS ?

HistoryTheory
Basic of X-ray scatteringStructural Information obtained by SAXS

Experimental Methods

X-ray Optics

X-ray Detectors

Advanced SAXSMicrobeam, GI-SAXS, USAXS, XPCS etc...
44

Emittance -- Product of size and divergence of beam

beam divergence
Ge Bent Monochromator
SamplePF BL-15A

Storage-Ring
2.5 GeV
250-400 mAd[photons/(smrad2mm20.1% rel.bandwidth)]

Focal Point

Bent Mirror
X-ray CCD Detector

beam sizedS
PF BL-10C

SAXS needs to use a low divergence and small beam
High brilliance beam is required !

4546

Good PointPSPC•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-

CCD
with Image
Intensifier

Fibertapered
CCD
47

Drawback

limitation

resolved

48

ImageIntensifier

Lens
systemVacuumCooledCCD

Coolingwater
In OutAnalog I/F

ReadoutUnite-Driver

Control Unit&14 bit A/De-Be
RbCsSbCsI:NaRS-232C

Phosphor screen
CCD Thermoelectro(Zn, Cd)S:Ag
coolerGate High-voltage
power-supply

Digital I/FPentium Pro
200MHz256MB Memory

Dataacquisition
system

IC-PCI includedIntroductionWhat’s SAXS ?

History

Theory

Basic of X-ray scatteringStructural Information obtained by SAXS

Experimental Methods
X-rray Optics

X-ray Detectors

Advanced SAXSMicrobeam, GI-SAXS, USAXS, XPCS etc...

Monitor49

Microbeam X-ray

50

Collab. with Kao ltd.

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

GI-SAXS- surface, interface,thin films

SAXS

Single FiberXPCS(30 - 100 m)- structural fluctuation- dynamics

African

Caucasian

Asian
X-ray Microbeam( 5 m x 5 m)

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

Relationship between macroscopic form
and nano structure?

- anisotropic structure
51

Local observation with an X-ray microbeam

Scattering pattern

1D intensity profile

Real space structure
DiameterFiber axis
-helix
SEM 像
Matrix
CuticleH. Ito et al., Textile Res. J. 54,397-402 (1986).

scattering intensity [a.u.]outerIF-IF Distance
Fibre Axis

（ globular protein ）Cortex0˚

R. D. B. Fraser et al., Proc. Int. Wool
Text. Res. Conf., Tokyo, II, 37, (1985)
partially changed.

Relationship between IF distribution
and hair curlness?

Diffraction peak originating from IF

ABCD

<d>

Intermediate Filament - 10 nm

53

innerIF

Coiled-coil dimer

InnerABCD
Outer
scattering vector s = 2 sin / (Å-1)

Difference in diffraction intensity--> Structural difference in cortex.
55

IF Tilt Angle
FWHM180˚090180IF54

Stained with Methylene blue dye

Distance: <d>para
Ortho


Diameter: 2

Outer side ofﬁber curve







Measuring Position

Optical microscope

Inner side ofﬁber curve

Microfibril

100 nmDiameter of Intermediate ﬁlament (IF) is almost constantDistance between IFs increases from outer to inner sides.

Nanocomposite

58

Rubber filled with spherical silica
54

TEM imageStress [MPa]Diameter or Distance (Å)

Distribution of Intermediate Filament (IF)3210sample- 160 mdetector050

100strain [%]150200elongation
Scattering pattern also shows hysteresis.
59

Y. Shinohara et al., J. Appl. Cryst., 40, s397 (2007).

60      
Sample：monodispersive silica spheres in rubber

x=0%

S(q): structure factor

F(q): form factor
Dave = 282.9 nm

0%

50%

100%

150%

I(q): intensity

corresponding to distances between silica particles

50 %

100 %

150 %

200 %

250 %

Y. Shinohara et al., J. Appl. Cryst., 40, s397 (2007).

Courtesy to Dr.Hagita & Prof. Arai

6162

Visualization of structure change of ﬁllers during elongation ofrubber by using SAXS and RMC
10 5０→150% : elongation process

Size of agglomerate

Scattering Intensity

10 4

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

10 3
largerstructure

10 2

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

10 1
10 0

Size of aggregate

10 -1
10 -2
non-uniformity increases along the elongation direction.

24

6 80.001

24

6 8-1

q /Å0.01

24

6 80.164Scattering Intensity

Measurement of fluctuation of X-ray scattering intensity--> Structural fluctuation in sample
coherent x-rayspeckle pattern    

I(q, 0)I∗ (q, τ)
g (q, τ) =
I(q)2(2)nano-particles in rubber

relaxation time in system
Autocorrelation

Autocorrelation

43100
150
Time /s

200

1.020
1.010

2146 8

210Time /sec

46 8100

WAXSt/s

103

10SAXS
10speckle pattern

010-150100
150Time /s

200

fluctuation ofscattering intensity

1.030
1.020
1.010
1.000
12246 8

210Time /sec

46 81002Dynamics of Filler in Rubber
66

65

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

1.000
50

3

Dependence of dynamics on...

1.030

504

Time-resolved SAXS with coherent X-rayT

Fluctuation of intensity
Scattering Intensity

100nm

Autocorrelation 5USAXS
10-2

10-3

10-410-5q/nm-1E /eV

10X-ray Photon Correlation
Spectroscopy

010Photon Correlation
10-6"cold"

Neutron Scattering
10-910NSEInelastic X-ray Scattering
10-1

1001011010-910210310-124105



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

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

L. A. Feigin and D. A. Svergun (1987) “Structure Analysis by Small Angle X-rray
and Neutron Scattering” Plenum Press.

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

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

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

-6

10-3

Brillouin
Raman

TOF

10-15

1010BS-12

Light Scattering

10-3-150r/nm68