X School on Synchrotron Radiation

“Small Angle X-ray Scattering”S.A.X.S.

STEFANO POLIZZI

DIP.CHIMICA FISICA

UNIVERSITÀ CA’ FOSCARI VENEZIA

X-Ray beam

Slits

Sample

Transmitted X-Ray

Beam stop

Detector

S.A.X.S.

W.A.X.S. (=XRD)

Why small angles?Why small angles?

All scattering phenomena are ruled by a reciprocal law through a Fourier transform: the larger the irradiated object, the smaller the scattering angle(if the wavelength of the incident radiation is constant)

As we approach the zero angle (the incoming direction in a transmission experiment), the dimensions we are exploring grow increasingly large.

λ [Å] 1° 0.1°0.15 nm (CuKα)α)α)α)

8 Κ8 Κ8 Κ8 ΚeV4.4 nm 44 nm

0.23 nm (CrKα)α)α)α)5.4 Κ5.4 Κ5.4 Κ5.4 ΚeV

6.8 nm 68 nm

……………… ……………….. ……………….400 nm (Visibile))))

3 3 3 3 eV11µm 110µm

How small is “small”?

So, if one is able to measure scattered intensity below 1°°°°from the incoming direction, one has a way to

investigate a range which spans from the atomic/molecular resolution of XRD to that of an

optical microscope.

Such dimensions are also called colloidal dimensions

Convolution

The convolution theorem:The convolution theorem:

The Fourier transform of the convolution of two functions is…

…the product of the Fourier transforms of the two functions (and viceversa)

?

Another example

Convolution

z(r )= ΣlΣmΣnδ(r -r lmn)ρc(r )CONVOLUTION

1/VcF(s)

σ(r )1 for |x|<L0 for |x|>L

z(r )= ΣlΣmΣnδ(r -r lmn)

Σ(s)

Z(s)= ΣlΣmΣnδ(s-slmn) CONVOLUZIONE

0.16 0.18 0.19 0.20 0.21 0.23 0.24 0.25 0.26 0.28

<D>=2π/∆h=1/∆s

200

111ZrO

2

h=4ππππsin( θθθθ )/λλλλ nm-1

XRP(owder)D: 1D section

Reciprocity of the Fourier Transform (2D)Reciprocity of the Fourier Transform (2D)

FTFTFTFTFTFTFTFT FTFTFTFT

Reciprocity of the Fourier Transform (1D)Reciprocity of the Fourier Transform (1D)

FTFTFTFT

FTFTFTFT FTFTFTFT

|F000(h)|2 | Σ(h)|2

S.A.X.S. W.A.X.S.

[|Fhkl(h)|2 | ΣhΣkΣl Σ(h-hhkl)|2]hkl≠000

<ρ2> | Σ(h)|2

0.2 0.3 0.3 0.4 0.4 0.5 0.6 0.60.0

0.2

0.4

0.6

0.8

1.0

1.2

222

311

220

200

111

h=4πsin(θ)/λ (nm-1)

Pd cuboctahedric cluster:Pd cuboctahedric cluster:Average diameter: 5 nmAverage diameter: 5 nm11 shells; 5083 atoms11 shells; 5083 atoms

nm

nmm

n mn rh

rhffhI

)sin()( ∑∑=

0.00 0.02 0.04 0.06

200

400

600

800

1000

inte

nsità

000

s [Å -1]

nm

nmm

n mn rh

rhffhI

)sin()( ∑∑=

2θ = 1.8° (λ=0.154 nm)

Pd cuboctahedric cluster:Pd cuboctahedric cluster:Average diameter: 5 nmAverage diameter: 5 nm11 shells; 5083 atoms11 shells; 5083 atoms

h [nm-1]

0.03 0.04 0.05

2

4

6

8

0.00 0.020

200

400

600

800

1000

inte

nsity

h[nm-1]

cubeoctaeder sphere

With low resolution the scatteringof the cubeoctaedric cluster is very similar to that of a sphere with radius 2.6 nm which contains the

same number of electrons

1E-3 0.01

inte

nsità

SFERA CLUSTER

s [ Å -1 ]

In logaritmic scale

h [nm-1]

( )matrixparticle

VI

ρρρρ

−=∆=∆=

electrons) ofnumber ()0( 222

( ) ( ) ( ) ( )( )

2

3

22 cossin3

−∆=hr

hrhrhrVhI ρ

SPHERE with Radius r

The electromagnetic radiation interacts with electrons. So, whatcauses small angle scattering are variations in the electron densityvariations in the electron densityof the irradiated matter on a scale which depends on the wavelenght of

the incoming radiation and on the scattering angle:

Dimensions of what?Dimensions of what?

This implies that what gives rise to small angle scattering are:

Amorphopus and/or crystalline particlesPores

BubblesCrazesetc …..

inside a homogeneous matrix

Structure 1 Structure 2

ρρρρ1111

ρρρρ2222

The two structures generate the same scattering:I(h)∝∝∝∝(∆ρ∆ρ∆ρ∆ρ)2 [Babinet’s principle]

h1

h2

Homogeneous particle (a portion of matter with a constant electronic density) dispersed in a matrix (a medium with a different electronic

density), e.g. a macromolecule in a solvent, a crystal phase-separated in a glass by thermal treatment, pores in a porous material.

2

3

222

0 3),,(3

4

−

∆=x

xcosxxsinabbaI πρh

22

221

2 hbhax +=

a

b

-0 .2 -0 .1 0 .0 0 .1 0 .21 E -9

1 E -8

1 E -7

1 E -6

1 E -5

1 E -4

1 E -3

0 .0 1

0 .1

1

1 0

h = 4 ππππs in ( θθθθ )/λλλλ

-0.2 -0.1 0.0 0.1 0.2

0.03125

0.0625

0.125

0.25

0.5

1

h

Different dimensions Different dimensions -- Same orientationSame orientation

I(h,a,b) = D(a,b)I0(h,a,b)dadb∫∫

Different dimensionsDifferent dimensionsIsotropic orientationIsotropic orientation

Small angles approximationSmall angles approximation

For dilutes systemsof particels at very low angles:

++−= ...)()(3

11)0()( 42

GG hROhRIhI

where Rg is the gyration radius.

This approximation allows one to easily determine particles dimensions

R g2 = 3

5r 2

R g2 = 3

5( a 2 + b 2 + c 2 )

R g2 = L 2

12+ r 2

2

scoordinatecenter -mass

)(

)(

)()( 22

2

=

−=

−−=

∫

∫

∫=

cm

V

cm

ρ

V

V

cmcm

g V

d

d

d

R

r

rrr

rr

rrrrr

costρ

ρ

What is the gyration radius?

r

ab

c

rL

Two approximations with the same first-order expansion series :

Zimmeq. 3/1

)0()(

Guinier eq. )0()(

220

3

22

0

hR

IhI

eIhI

gh

hgR

h

+≅

≅

→

−

→

Both approximations may be easily linearized by suitable plots

Zimm

Guinier

)0(3)0(

1

0)(

1

3)0(ln

0)(ln

22

22

I

hR

IhhI

hRI

hhI

g

g

+→≅

−→≅

0.0 5.0x10 -4 1.0x10 -3

e -2

e -1

e0

e1

e2

e3

e4

e5

e6

e7

Inte

nsità

h^2

cubo-ottaedro Guinier

Guinier approximation: 3

22

0)0()(

hgR

heIhI

−

→≅

0.1 1

3 6 9 12 15 18 21

h [nm -1]

2

1

<R>=6nm

nm

POLIDISPERSITY EFFECT on the scattering intensity

POLYDISPERSITY EFFECT on the gyration radiusPOLYDISPERSITY EFFECT on the gyration radius

For a polydispersion the radius of gyration is a weighed average wich largely overestimates the contribution of the

lager particles

RG2 = 3

5

R8

0

∞

∫ N(R)dR

R6 N(R)dR0

∞

∫

N(R) number of particles with radius between R and R+dR

0.0 0.1 0.2 0.3 0.4 0.5

e3

e4

e5

e6

e7

e8

e9

e10

6 12 18 24 30

h2[nm -2]

<R>=6nm

nm

2

3

5Gapp RR =

RG= 4.6 nmRapp= 6.0nm

RG= 6.5 nmRapp= 8.4 nm

RG= 19.3 nmRapp= 24.9nm

POLYDISPERSITY EFFECT POLYDISPERSITY EFFECT on the gyration radiuson the gyration radius

Asintoptic trend: the Porod’s Law

ShhI

hh

ShI

h∝

∞→∆∝

∞→

4

4

2

)(

2

)(ρπ

Holds for a “regular” separation interface.

For two phase systems S represents the surface area of separation between the tho phases

1E-3 0.01 0.1

0.01

0.1

1

10

100

1000

inte

nsity

h

CUBO-OTTAEDRO POROD GUINIER

GUINIER POROD

0.0 0.2 0.4 0.6 0.8 0.01 0.1 1

RG=2.1

RG=8.2

inte

nsity

a.u

.

h2(nm-2)

slope -4

h (nm-1)

Two groups of particles with distinct dimensions

GUINIER POROD

Q = 2π 2 V (∆ρ)2ϕ(1−ϕ) = I h( )0

∞

∫ h2dh

Other important equations: the Invariant

I 1(h) ≠≠≠≠ I 2(h) Q1 = Q2

π ϕ (1−ϕ) I(h) h4

I h( )0

∞

∫ h2dh

h →∞ → constant= S

V

Measure of the specific surface

1 2

V

S

V

S 21 >

12

Log(I(h))

Log(h)

2

1

h-4

In this case the total intensity is not the simple sum of the intensities of the individual

scattering particles

I(h) ≠ N I0(h)

I(h) ≠ P(r)∫ I0(h,r) dr

Interference effects between partciles must be taken into account

N Equal particles

Different particles

NON DILUTED SYSTEMS

0.00 0.02 0.04 0.06 0.08 0.100

1

2

3

0 3 6 9 12 15 18

h [nm-1]

N(r)

r [nm]

Rint<r>=6 nm<Rint>=2<r>=12 nm

Form factor P(h)

Structure factor S(h)

Scattering intensityI(h)=P(h) S(h)

int2

2

R

π

Measurements are usually carried out in transmission

The ideal beam is

� Monochromatic� Point-like� Well collimated

LOW INTESITY

� Increase the sample-detector distance� Increase λ

“From the experimental point of view […], small-angle scattering appears to have reached a steady value. The apparatus for small-angle scattering will certainly be continuously improved but no major change can be foreseen, unless the power of X-ray sources is increased by a factor 10 or 100, which is rather unlikely”.

GUINIER: 1969

Grenoble synchrotron radiation:1012 more brilliant than a conventional source

This opens new frontiers

� time-resolved measurements

� 2D- Detectors

� Local measurements (microdiffusion)

� Anomalous Scattering

The starting material is photochromic glass containing Ag(Cl,Br) crystallites

�Heating to 725 °°°°C ⇒⇒⇒⇒ Ag(Cl,Br) droplets

�Drawing at T>Tsoft ⇒⇒⇒⇒ cigar-like Ag(Cl,Br) particles

�Reduction at 430°°°°C in H2 ⇒⇒⇒⇒ cigar-like Ag particles

Example 1: Polarizing glasses S. Polizzi et al,

J. Appl. Cryst., 30, 487 (1997);J. Non-Cryst. Solids 232----234, 147 (1998)

VETRI POLARIZZATORI

200nm

ULTRA-SAXS (HASYLAB-DESY): Sample-detector distance: 12 m ; λλλλ=0.124nm

I(h,a,b) = D(a,b)I0(h,a,b)dadb∫∫

a = η r

b = r

η

r

ab

stretching

I(h,a,b) = D(r)I0(h,η r,r

η)dr∫

η (r) =1+ ηlim 1− exp −(r /rlim )m[ ]{ }

0.00

0.02

0.04

0.06

0 10 20 30 40 50 60

5

10

15

ηη ηη(R)

D(R

)

R [nm]

0 200 400 6000.000

0.004

0.007

0.011

0.014D

(2a)

length (2a) [nm]

0 10 20 300.00

0.01

0.02

0.03

0.04

0.05

D(2

b)

width(2b) [nm]

Length distribution

Width distribution

EXAMPLE 2:Aggregation of colloidal systems:

SULPHATE ZIRCONIA SOL-GEL

FRACTALITY

An object is called “fractals” when it shows a scale-invariance in a particular length range

M∝R Df

Df =1, 2, 3 for euclidean objects1≤≤≤≤Df<3 for fractal objects

M=Object MassR= Object Radius

The fractal dimension of a surface ds

comes out to be

S∝RDs

S=Surfaceds=2 for non fractal “regular” surfaces2<ds<3 for fractal surfaces

One finds out that such trends translate in the reciprocal space so that the small angle scattering fractal dimensions are obtained by

I(h)∝h -Df

I(h)∝ h Ds-6

1E-3 0.01 0.1 11E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1

10

h-Df

h-6+Ds

Raggregato

1/rmonomero

1/Raggregato

inte

nsity

h

I(h) = Nvo2 ∆ρ( )2

imon (h)

I(0) = Nvo2 ∆ρ( )2

imon (0) =1

I(h) = N

kk2vo

2 ∆ρ( )2iagg (h)

I(0) = k Nvo2 ∆ρ( )2

∝ k ∝ mass

iagg (0) =1

N identical colloidal particles (monomers) with volume vo

M =N/k aggregates of k particles

Thus one can measure the aggreates mass [I(0)]And dimension [Rg ] without any assumption on

their structure

I(0) ∝ M∝R Df

Measuring the scattering as a function of time, it is possible to calculate Df and thus

determine the growth mechanism

1

0.01

0.1

1

h (nm-1)

min

454

806683667

354

134

226

61

25

2

inte

nsity

(a.

u)

78.1gRM ∝

gRM ∝

DIFFUSION LIMITED CLUSTER AGGREGATION

P.Riello et al. J.Phys.Chem. 107, 15 (2003) 3390

1 10

0.1

1

RG(nm)

slope 1.78(6)

slope 0.98(6)

I(0)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.00

0.02

0.04

0.06

0.08

0 20 40 60 80 600 800 10000.2

0.4

0.6

2

4

6

(b)I(0)

slope 0.0270±0.0005

<L> (nm)

Io/I

(a)

0.528±0.003

Radius Attenuation I/I

o

Ro (nm)

min

RoRo

L

EXAMPLE 3: Nanostructure of Pd/SiO2 catalysts

A. Benedetti et al. J. Catal. 171, 345 (1997)S Polizzi et al P.C.C.P., 2001, 3, 4614,4619

J. Synchrotron Rad. (2002). 9, 65±70

Catalysis is a surface phenomenon

� Efficient use of expensive metals

� Different electronic structure

� Increase of catalytical activity and selectivity

WHY NANOPARTICLES ?WHY NANOPARTICLES ?

Small particles→ high surface/volume ratio

20 40 60 80 100 1200

20

40

60

80

100

120

2θ°

Pd 6%

Pd 3%

Inte

nsity

[cou

nts/

s]

Wide-Angle X-ray Scattering

35 36 37 38 39 40 41 42 4312

16

20

Pd 3%

Peak intensities too small Peak intensities too small compared to those of 6 wt%compared to those of 6 wt%

Hint for very small particles (clusters)Hint for very small particles (clusters)

Particles smaller than 4Particles smaller than 4--5 nm5 nm

Very broad peak with superVery broad peak with super--lorentzian lorentzian shapeshape

0 5 10 15 20 25 30 35 40 45 500.0

0.1

0.2

0.3

0.4

0.5

0.0

0.1

0.2

0.3

0.4

0.5

Pd 3%

Pd 3%

Pd 6%

D

istr

ibut

ion

of p

artic

les

Diameter [nm]

23.0 23.2 23.4 23.6 23.8 24.0 24.2 24.4 24.6-12

-10

-8

-6

-4

-2

0

2

4

E0

E1

E2

E3

E4

f '

f "

f', f"

(el

ectr

ons)

Energy (keV)

Anomalous Small-Angle X-ray Scattering

f = f0 + f ´ (E) + f ́ ´(E)

0.1 1

101

102

103

104

Inte

nsity

[a.u

.]

h [nm-1]

E1

E5

1570

1580

1590

1600

1610

1620

1630

1640

1650

2.252712.255 2.26 2.265 2.27 2.275 2.28 2.285 2.29

46

47

48

49

50

Both samples show a double distribution of particles

The total surface area is the same for both

samples

0.1 1102

103

104

105

106

Pd 3% Pd 6%

h [nm-1]

inte

nsity

[el2 /n

m3 ]

0 5 10 15 20 25 300.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Pd 6%

Pd 3%

Dis

trib

utio

n of

par

ticle

s

Diameter [nm]

Smaller particles are

80% in the 3 wt% sample80% in the 3 wt% sample

56% in the 6 wt% sample56% in the 6 wt% sample

The size of the clusters increases with increasing metal content

EXAMPLE 4: SiO2-PEG Hybrid materials

Obtained by hidrolysis of the precursor(OEt)3Si-(PEG)-Si(OEt)3

Suitably doped they can be:

•Ionic conductors•Photocromic materials• Luminescent materials•……..

Karim Dahmouche et al. J. Phys. Chem. B 1999, 103, 4937-4942

Microstructure determine by SAXS analysis

EFFECT OF PEG MOLECULAR WEIGTH Mw

1900 g/mole

800 g/mole

500 g/mole

200 g/mole

hmax

h[Å -1]

Average distance of SiO2 clusters in the polimeric matrix:

max

2

hd s

π=

( )3

1

ws Md ≅

( )

3223

2

0

22

3

4 )(2

3

4

radius with particles spherical of system aFor

1)-(1 systems diluitedFor

clusters SiO2by occupied volume

)1()(2

cc

c

RNQrRNV

RN

V

dhhhIVQ

πρππϕ

ϕϕ

ϕϕρπ

∆=⇒=

≅=

=−∆= ∫∞

INVARIANT

224

4

2

42)(

2

)(

:particles spherical of system aFor

cRhhI

hh

ShI

limh

N

πρπ

ρπ

∆=

∞→∆∝

∞→

POROD’S LAW

cRhhI

Q

limh

3)( 4

π=

∞→

From whichRc can be calculated:

ds=20-60Å

Rc=3-6 Å

EXAMPLE 5:

Grazing Incidence Small-Angle X-ray Scattering (GISAXS)of Cu-Ni alloys clusters obtained by implantation in a glassy matrix

E. Cattaruzza et al. J. Appl. Cryst. (2000). 33, 740-743,

In-detph view Plan view

Courtesy of: M. Buljan et al.Vacuum 71 (2003)65-70

Grazing Incidence Small-Angle X-ray Scattering (GISAXS)

0.0 0.2 0.4 0.6 0.80.0

0.2

0.4

0.6

0.8

1.0

Materiale SiO2

Enegia Rx: 8Kev, (0.154nm)

Rifl

ettiv

ità

angolo di incidenza (deg)

( ) ( ) ( )( )

b)a,on (dependenton distributi radia ),,(

clustersbetween radiusn Interactio

Yevich)-(Percus PYfactor Structure ),(

cossin3

3

4 ),(

),,(),(),()(

int

int

2

3

23

0

int

=•==•

=•

−

∆=Φ•

Φ= ∫∞

baRP

RR

RhS

hr

hrhrhrRRh

dRbaRPRhSRhKhI

α

ρπ

RR α=int

intR

R

<R> = 21 Åα = 1.2

Atoms/cm3= 1.5 1022 (GISAXS)2.0 1022 (RBS)

Volume fraction occupied by particles = 0.16

0.04 0.06 0.08 0.10 0.12 0.14

exp. data

LMA fit

scat

tere

d in

tens

ity (

arb.

u.)

q(Å-1)

1

3

2

0

EXAMPLE 6: Low resolution

structure of macromolecules

Dmitri I. Svergun Michel H. J.Current Opinion in Structural Biology 2002, 12, 654-660

Svergun D.I. J. Appl. Cryst.

(1997). 30, 792-797;

Dmitri I. Svergun Michel H. J.Current Opinion in Structural Biology 2002, 12, 654-660

,

B

Yeats Hexokinase: the monomere structure is known, but the biologicaly active form is a dimer, whose quaternary structure in solution is uncertain

A B

A

Peter Fratzl

J. Appl. Cryst. (2003). 36, 397±404

ESEMPIO 7:

MICRODIFFUSION