October 2, 2011: Cheiron School 2011 @ SPring-8

Table of Contents!

Small-Angle X-ray Scattering
Basics & Applications!

!

Yoshiyuki Amemiya and Yuya Shinohara
Graduate School of Frontier Sciences,!

The University of Tokyo

Introduction!

What’s SAXS ?

!

History

Theory!

Basic of X-ray scattering!

Structural Information obtained by SAXS

Experimental Methods!

X-ray Optics!

X-ray Detectors

Advanced SAXS!

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

Interference of secondary waves

What’s Small-Angle X-ray Scattering ?
scattered

x-ray

2θ

transmitted
samplearea detector
Imaging Plate,
CCD etc.

Incident X-ray

beam stop

Bragg’s law:

Each electron in marerials vibrates and emits secondary spherical wave

When there are two electrons,

small angle

large structure(1 - 100 nm)

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

interference between secondary waves from electrons•

when a distance between electrons is large

when a distance between electrons is small--> scattering at large angle is intensiﬁed--> scattering at small angle is intensiﬁed --> SAXS
4History of SAXS (< 1936)
Krishnamurty (1930)
Hendricks (1932)
Mark (1932)

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

Observation of scattering

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

from powders, fibers, and colloidal
dispersions

Warren (1936)
O. Kratky (1938, 1942, 1962)
G. Porod (1942, 1960, 1961)
Description of dense systems of
colloidal particles, micelles, andfibers.
Macromolecules in solution.

carbon black

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

Single crystals of AlCu hardened alloy

5

Application of SAXS

Hemoglobin
6

Table of Contents!

!

Typical SAXS image
gelcourtesy to Dr. I.L.TorrianiProteins in solution (Dr. Svergun, EMBL)!

!

100 nmIntroduction!

What’s SAXS ?

!

History

Theory!

Basic of X-ray scattering!

Structural Information obtained by SAXS

Experimental Methods!

X-ray Optics!

X-ray Detectors

Advanced SAXS!

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

Nanocomposite
78

Real space and Reciprocal Space

Scattered X-ray(wavenumber: ks)

Real Space

Incident X-ray

Scattering vector

ks(wavenmber: ki)r2θ

q = ks - kiρ(r) : electron densitykiAutocorrelation

Sampleq = |q|= 4π sin θ /λAmplitude of scattered X-ray
Fourier transform of electron density
Scattering intensity per unit volume:
Intensive variables don’t depend on the system size (volume).
Extensive variables depend on the system size (volume).

Electron Density

9

Reciprocal Space
Fourier Trans.

Scattering amplitude
A(q)

ρ(r)
Inv. Fourier Trans.

I(q)

γ(r)
Autocorrelation
Function

Scattering Intensity= Patterson Function per volume

10!(r) & "(r) of lamellar structure

Autocorrelation Function & Scattering Intensity

ρeAutocorrelation function of electron density per unit volumeLideal orderingdcγ(∞) = "ρ#Qρc - ρa

dc
L
ρe

r

γ (r)
Q

asymptotic behavior of the autocorrelation function

γ(0) = !ρ "γ (r)Patterson Function(Debye & Bueche 1949)

2

Squaring

Basic of X-ray scattering

Long period changes.

2ρc - ρa

dc
r
ρe

γ (r)
Q

Scattering Intensity : Fourier Transform of autocorrelation function
Thickness of crystal changes.

ρc - ρadc
rreal space
11

autocorrelation
12

Normalized Autocorrelation Function:
Introducing the relative electron density,

Invariant Q

η(r) , asInvariant:doesn’t depend on the structure

Introducing the normalized autocorrelation function, γ0 (r) , as!
1
!η2 " γ0 (r) =η(r! )η(r + r! )dr!
V V
then,

γ0 (r) =γ(r) − "ρ#2
"η2 #

where

!

（

γ0 (0) = 1 γ0 (∞) = 0）

Omitted.

Parseval!s theorem

Parseval’s theorem
Fourier Trans.

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

Not observable.
13

14

SAXS of particulate system

Information obtained from SAXS

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 system
Colloids, Globular proteins, etc...

Scattering Intensity I(q)!

2. Correlation length of inhomogeneous structure!

Polymer chain, two-phase system etc.

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

Crystalline polymer

15

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 q
q = 4π sin θ /λ16

Spherical sample

Homogeneous sphere
2

ρ (r)

I (q)

Fourier Transform& squared
r
0

R

q
2

isotropic scattering
courtesy to Dr. I.L.Torriani

17

Homogeneous elipsiod

Size distribution
Larger distribution

Random orientation
Scattering Intensity

Fixed particle

anisotropic scattering

18

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

Based on Gaussian distribution

When the form has distribution,
fringes are missed.

isotropic scattering0
courtesy to Dr. I.L.Torriani

19

1

2

3
-1
q/Å

4

-2
5 6x1020Structure Factor & Form Factor
Scattering Intensity I(q)Radius of Gyration -- Guinier Plot

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

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

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

Structure Factor Form Factor

Inter-atomic structure(WAXD)

Size of particle

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

Scattering angle 2θ or Scattering vector q
q = 4π sin θ /λintra-particle
structureinter-particle structure

Separation 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

21

22

Scattering from Inhomogeneous Structure

Two-phase system

Autocorrelation Function

Phase 1: !1, volume fraction # Phase 2: !2 volume fraction 1 - #Electron Density

Scattering Intensity! "rρ̃(r) = exp −ξ

two phase system
Autocorrelationρ̃(r) =

Debye-BuecheI(q) =

1(1 + ξ 2 q 2 )2Fourier trans.

!
"
ξ
r
exp −
r
ξ

Ornstein-Zernike

I(q) =

1
1 + ξ2 q2

Babinet’s principle

polymer chain etc.
23

Two complementary structures produce the same scattering.

24

Two-phase system -- cont.

Porod’s law

Averaged square fluctuation of electron density

For a sharp interface, the scattered intensity decreases as

q-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.
25

Pair Distance Distribution Function: PDDF

26

I(q) & P(r) of Spherical particle

Scattering intensity:(when isotropic)
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 of all intra-particle distances

pairs of volume
elements i-j

27

courtesy to Dr. I.L.Torriani

28

I(q) & P(r) of Cylindrical particle

courtesy to Dr. I.L.Torriani

I(q) & P(r) of Flat particle

29

I(q) & P(r) of Ellipsoids

courtesy to Dr. I.L.Torriani

30

I(q) & P(r) of Two ellipsoid = dimer

ProlateOblatecourtesy to Dr. I.L.Torriani

31

courtesy to Dr. I.L.Torriani

32

Diffraction from Periodic Structure

Laue Function

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 ~ 2!/N!

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

!!
!!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 intensity!

Wide FWHM

N = 50low S/N

FWHM --> Size of crystal

N = 100

N = 10

Crystal size --> Intensity & FWHM of diffraction
33

Imperfection of crystal (2D)34

Imperfection of crystal
Imperfection of 1st kind

Imperfection of 2nd kind

Autocorrelation

Imperfection of 1st kind

Imperfection of 2nd kind

Thermal fluctuation etc.

in the case of soft matter35

36

Imperfection of lattice (1D)

Diffraction from lattice-structure
A(q)

(a)
Perfect lattice

Diffraction

rj(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)3738Imperfection of 2nd kind

Table of Contents

4!

|Z(q)|2/N

3
2

!

1
Paracrystal theory

!

0
0

1

2

3

4

5

6

q / (2d!!

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

39!

Introduction!

What’s SAXS ?

!

History

Theory!

Basic of X-ray scattering!

Structural Information obtained by SAXS

Experimental Methods!

X-ray Optics!

X-ray Detectors

Advanced SAXS!

Microbeam, GI-SAXS, USAXS, XPCS etc...
40

X-ray Source for SAXSEmittance -- Product of size and divergence of beam

X-ray Optics for SAXS
beam divergence
Ge Bent Monochromator

PF BL-15A

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

SampleStorage-Ring
Bent Mirror

Focal Point

X-ray CCD Detector

beam sizedS
PF BL-10C

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

41

42

Slits for SAXS

Detectors for SAXSGood Point
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-

CCD

with Image
Intensifier

Fibertapered
CCD
43

Drawback

limitation

resolved44

X-ray CCD detector with Image Intensifier
Image
Intensifier

Lens
systemVacuumCooledCCD

Coolingwater
In OutAnalog I/F

Readout
Unite-

DriverTable of Contents!

Control Unit&
14 bit A/D

!

e-

Be
CsI:NaRbCsSb

RS-232C

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

Digital I/FPentium Pro
200MHz256MB MemoryDataacquisition
system

IC-PCI included!

Introduction!

What’s SAXS ?

!

History

Theory!

Basic of X-ray scattering!

Structural Information obtained by SAXS

Experimental Methods!

X-ray Optics!

X-ray Detectors

Advanced SAXS!

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

45

Advanced SAXSMicrobeam X-ray

46

Application of paracrystal theory
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 Fiber

XPCS(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 microscopic structure?

- anisotropic structure
47

Local observation with an X-ray microbeam

48

Internal structure of wool

Structure of Intermediate Filament
Scattering pattern

1D intensity profile

Real space structure
DiameterFiber axis'-helix

SEM 像

Coiled-coil dimer

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

MatrixFibre 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 intensity profiles

outerscattering intensity [a.u.]

Diffraction peak originating from IF

ABCDIF-IF DistanceIntermediate Filament - 10 nm

49

inner

IF

<d>

Inner
A
B
C
D
Outer
scattering vector s = 2 sin(/) (Å-1)

Difference in diffraction intensity--> Structural difference in cortex.
51

180˚

IF Tilt Angle
FWHM
0

90

180

IF

50

Diameter or Distance (Å)

Distribution of Intermediate Filament (IF)USAXS for nano-composite in rubber
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 nm* Diameter of Intermediate ﬁlament (IF) is almost constant* Distance between IFs increases from outer to inner sides.

Nanocomposite

54

USAXS patterns from elongated rubber

USAXS using medium-length beamline

Rubber filled with spherical silica
5

TEM imageStress [MPa]

4

3

2

1

0

sample- 160 mdetector

0

50

100

strain [%]

150

200

elongation
Scattering pattern also shows hysteresis.
55

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

56

!"#$%$&'()!(*!!&%+,&+%"!*$,&(%!!-./!

Analysis by RMC (Reverse Monte Carlo)
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).

Visualization of structure change of fillers during elongation ofrubber by using SAXS and RMC

Courtesy to Dr.Hagita & Prof. Arai

57

58

Structural information from USAXS
10 5

０→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.

2

4

6 8

0.001

2

4

6 8

-1

q /Å

0.01

2

4

6 8

0.1
60

Dynamics of nanoparticles observed with XPCS

X-ray Photon Correlation Spectroscopy: XPCS!"#$%&'()&*+,(-'$,,./&#*((0(!)-(1coherent x-ray

0(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+-#'

#%

$,-,(.)%'+/0$(
121.$&1+"&!""""""""""""""

!(2)
%"

$

"#$

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

62&/'+"&

Scattering Intensity!

) + '#0& (*!

100
150
Time /sspeckle pattern- Volume fraction of nano-particles- Vulcanization (cross-linking)- Type of nano-particles- Temperature
etc.

1.020
1.010

+&#&&'0 -+:$1(+&6"%5)'+"&("&(1/)''$%$% *<&)5+/1

1.000

200

1

2

4

6 8

2

10
Time /sec

4

6 8

100

t/s

103

10

1

SAXS
10

0

10

-1

10

1.020
1.010
1.000
1

4

6 8

2

10
Time /sec

4

6 8

100

2

62
Dynamics of Filler in Rubber

Bibliographyq/nm-1

10-15"cold"

Neutron Scattering-9

NSE

Brillouin
Raman

TOF
Inelastic X-ray Scattering
10-1

100

!

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

10-12

!

O. Glatter and O. Kratky ed. (1982) “Small Angle X-ray Scattering” Academic
Press, London. out-of-print-9!

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

!

P. Lindner and Th. Zemb ed. (2002) “Neutron, X-ray and Light Scattering: Soft
Condensed 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.

10

10-6

BS

10-12

Light Scattering

Photon Correlation

10-6

-15

10-5

X-ray Photon Correlation
Spectroscopy

10-3

10

10-4

2

E /eV

100

10

10-3

200

1.030

2

USAXS
-2

100
150
Time /sfluctuation of
scattering intensity

61

WAXS

50

Dependence of dynamics on...

1.030

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

50

relaxation time in system'()*

Autocorrelation

'()*

3

!

@$0)<('+5$(((((&

3
0

%

1%(2!

)* ()*+,./)*+% ";&'$&1+'<()'(
15)00(4*$'$/'"%

40

nano-particles in rubber!

5Time-resolved SAXS with coherent X-ray

3/%$$&>1(1/)''$%+&-(5$*+25($:"0:$1?(
Fluctuation of intensity;&'$&1+'<( 1%(!2
1.$/80$(.)''$%&(602/'2)'$1)2'"/"%%$0)'+"&(
5%&!&$'

100nm

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

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

Scattering Intensity

Measurement of fluctuation of X-ray scattering intensity--> Structural fluctuation in sample

!

101

102

103

104

105

10

-3

10

0

r/nm
64