31.05. : Small-Angle X-ray Scattering (SAXS) 02.06. : …photon-science.desy.de/sites/site_photonscience/content/e62/e... · T-SAXS vs GISAXS - Easy measurement - Easy analysis -

Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

SoSe 2016, S. Roth

> 31.05. : Small-Angle X-ray Scattering (SAXS)

> 02.06. : Applications &

A short excursion into Polymeric materials

> 04.06. : Grazing incidence SAXS (GISAXS)

2 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

SoSe 2016, S. Roth

T-SAXS vs GISAXS

- Easy measurement

- Easy analysis

- In-plane information (qy,qz)

- Any possible scattering from

substrate

- Transparency of substrate

- High energy

Lee et al., Macromolecules, 38, 8991 (2005)

- Strong intensity

- Easy preparation of samples

- Full information (qx,qy,qz)

- Scattering from surface /

internal structure

- Scattering from reflected AND

transmitted beam

- Refraction effects (DWBA)

- Special setup

Si

qy

qx

qz z y

x

2q

Beamstop

af

2q x

y z

ai

qy

qx

qz


SoSe 2016, S. Roth

Aim

> To understand the structure – property relation of materials on

multiple length scales

- q-resolution

- Maximum q-value

- Beam size

- Real pieces

& materials

- Model systems

- Nanotechnology

http://news.thomasnet.com/companystory/

GE-Gas-Turbine-Technology-Selected-

for-Pearl-GTL-Project-in-Qatar-495497

Courtesy: R. Gilles (TUM)


SoSe 2016, S. Roth

Outline II - Today

> SAXS – Introduction

> Instrumentation

P03/MiNaXS @ PETRA III

> Bulk materials Transmission U/SAXS:

Porous materials

Ni-base superalloys

Droplet drying

Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter.

Eds: P. Lindner, Th. Zemb. North Holland Delta Series, Elsevier, Amsterdam (2002)

ISBN: 0-444-51122-9


SoSe 2016, S. Roth

Cross Section

> Differential cross section

> Scattering occurs due to density differences

𝑑𝜎 =𝐼

𝐼0𝐿2𝑑Ω

𝑑𝜎

𝑑Ω=𝐼

𝐼0𝐿2 ⇨

𝑑Σ

𝑑Ω=1

𝑉

𝑑σ

𝑑Ω

V = Sample volume

𝑞 = 𝑘𝑓-𝑘𝑖

2 if kk

𝑞 = 2

2𝜋

𝜆sin(𝜃)

dW

Q 2q

kf

ki

Detector → I

I0

2

L


SoSe 2016, S. Roth

WAX, SAX, GISAX

Source: Streumethoden zur Untersuchung kondensierter Materie

1996; ISBN 978-3-89336-180-9

SAXS WAXS

Limit Guinier Porod Bragg

Lo

g I

(q)

WAXS:

Crystal

structure

SAXS/GISAXS:

density fluctuations, precipitates

d / resolution [Å]

1 10

=1nm

100 1000 10000

=1µm 100000

=10µm

Log q 𝜆 = 2𝑑 sin 𝜃

𝜆 = 1.54Å > R~particles “radius“

> d~interatomic distance

> SAXS: q < 5°

d

R


SoSe 2016, S. Roth

Scattering Amplitude

> Interference in far field

> Phase difference:

> Scattering amplitude:

> Intensity:

Δ𝜑𝑖 = 𝑘𝑓 − 𝑘𝑖 ∙ 𝑟 𝑖 = 𝑞 ∙ 𝑟 𝑖

𝐴 𝑞 = 𝜌 𝑟 𝑒−𝑖𝑞𝑟 𝑑𝑉 = 𝜌 𝑟 𝑒−𝑖𝑞𝑟 𝑑3𝑟

𝐼 𝑞 = 1

𝑉𝐴 𝑞 2

q kf

q

q

ki

kf

2q


SoSe 2016, S. Roth

Form factor and structure factor: Fourier transform

Single particle: Fourier transformation

𝐴 𝑞 = 𝜌𝑃 𝑟 𝑒−𝑖𝑞𝑟 𝑑𝑉 = 𝜌𝑃 𝑟 𝑒

−𝑖𝑞𝑟 𝑑3𝑟

Particle distribution function G(r) Electron density distribution

𝜌 𝑟 = 𝜌𝑃 𝑟 𝑖𝑖

= 𝜌𝑃 𝑟 ′ 𝐺(𝑟 − 𝑟 ′)𝑑3𝑟′ = 𝜌𝑃 𝑟 ∗ 𝐺(𝑟 )

Scattering amplitudes of the whole arrangement

𝐴 𝑞 = 𝜌 𝑟 𝑒−𝑖𝑞𝑟 𝑑𝑉 = [𝜌𝑃 𝑟 ∗ 𝐺(𝑟 ) ]𝑒−𝑖𝑞𝑟 𝑑3𝑟

= 𝜌𝑃 𝑟 𝑒−𝑖𝑞𝑟 𝑑𝑉 ∙ 𝐺 𝑟 𝑒−𝑖𝑞𝑟 𝑑𝑉

Scattered Intensity

𝐼 𝑞 = 1

𝑉𝐴 𝑞 2 = 𝑃 𝑞 𝑆(𝑞 )

Form factor Structure factor

)(rp

i

ri


SoSe 2016, S. Roth

Two phase Model: Dilute systems

> Only form of particle relevant

> Matrix M, volume fraction F

Particles P , volume fraction (1-F)

Electron density: M,P=nM,P*fM,P

fM,P : atomic form factor („extension of the electron cloud“, resonances)

nM,P : number density of atoms

> Consider M,P as constant resp.

2R

ASAX

2R

2R

x >> R


SoSe 2016, S. Roth

Two phase Model

> Scattering amplitude:

𝐴 𝑞 = 𝜌 𝑟 𝑒−𝑖𝑞𝑟 𝑑3𝑟 = 𝜌𝑀 𝑟 𝑒−𝑖𝑞𝑟 𝑑3𝑟

Φ𝑉+ 𝜌𝑃 𝑟 𝑒

−𝑖𝑞𝑟 𝑑3𝑟 (1−Φ)𝑉

𝐴 𝑞 = 𝜌𝑀 − 𝜌𝑃 𝑒−𝑖𝑞𝑟 𝑑3𝑟

Φ𝑉

𝐴 𝑞 = Δ𝜌 𝑒−𝑖𝑞𝑟 𝑑3𝑟 Φ𝑉

> 𝐼 𝑞 = 1

𝑉𝐴 𝑞 2~∆𝜌2

> Porod Invariant Q (Porod, 1982):

Q = 𝐼 𝑞 𝑑3𝑞 = 4𝜋Φ(1 − Φ)Δ𝜌2

> Only dependent on density contrast D

Ableiten!

Mittelung <..> erklären S.25, S.51


SoSe 2016, S. Roth

Herleitung Porod-Invariante

> Siehe Handzettel und Übung

> Q-Berechnung Übung


SoSe 2016, S. Roth

Two phase Model – single particle approximation

> Amplitude: 𝐴 𝑞 = Δ𝜌 𝑒−𝑖𝑞𝑟 𝑑3𝑟 Φ𝑉

> Intensity: 𝐼 𝑞 = 1

𝑉𝐴 𝑞 2

> Closer look at I(q) for dilute systems: NP independent scatterers

> Incoherent sum of intensities:

𝐼 𝑞 ~𝑁𝑃𝑉𝑃2Δ𝜌2

1

𝑉𝑃 𝑒−𝑖𝑞𝑟

𝑉𝑃

𝑑3𝑟

2

2R 2R

nPfP

nMfM Δ𝜌

r

VP~R3

nMfM

=M

nPfP=

P


SoSe 2016, S. Roth

Two phase Model – single particle approximation

> Amplitude: 𝐴 𝑞 = Δ𝜌 𝑒−𝑖𝑞𝑟 𝑑3𝑟 Φ𝑉

> Intensity: 𝐼 𝑞 = 1

𝑉𝐴 𝑞 2

> Closer look at I(q) for dilute systems: NP independent scatterers

> Incoherent sum of intensities:

𝐼𝑚 𝑞 ~𝑁𝑃𝑉𝑃2Δ𝜌2

1

𝑉𝑃 𝑒−𝑖𝑞𝑟

𝑉𝑃

𝑑3𝑟

2

2

3)(

)cos()sin(3)(

qR

qRqRqRqP

- Form factor of a sphere of radius R - Isotropic scattering


SoSe 2016, S. Roth

Colloid: homogeneous sphere of radius R

A simple, but important calculation:

drddrerderqF rqi

R

lumeparticleVoV

rqi qq

)sin()()( 2

0

2

0 0

0

3

R iqriqriqr

R

drrqr

eedrddreqF

0

2

0

2)cos(

0 0

0 )sin(2)sin(2)( qqq q

RRR

drq

qr

q

qrr

qdrrqr

qqF

00

0

0

0

)cos()cos(4)sin(

22)(

30

3

2

0 )cos()sin(4

)sin()cos(4)(

qR

qRqRqRR

q

qR

q

qRR

qqF


SoSe 2016, S. Roth

Colloid:homogeneous sphere of radius R


SoSe 2016, S. Roth

Guinier radius

> Q → 0

> Homogenous sphere of radius R

> Radius of gyration:

replace homogenous sphere by shell of same moment of intertia: Rg

> 𝑅𝑔 =35 𝑅

> general form of Guinier law [Guinier (1955)]

> Independent of particle form

𝑃 𝑞 ~exp(−1

3𝑞2𝑅𝑔

2)

𝑃 𝑞 = 3sin 𝑞𝑅 − 𝑞𝑅𝑐𝑜𝑠(𝑞𝑅)

𝑞𝑅 3

2

~1 −1

5𝑞2𝑅2~exp(−

1

5𝑞2𝑅2) Ableiten


SoSe 2016, S. Roth

Guinier Approximation

)3

exp()(lim

2

222

0

g

q

RqVqI D

Radius of Gyration Rg

Monodisperse spheres of radius R: RRg 5/3Roth et al., Appl. Phys. Lett. 91, 091915 (2007)

2nm Colloids

domains


SoSe 2016, S. Roth

Porod´s law: large q

Scattered intensity:

2

30

3 )cos()sin(4~

qR

qRqRqRR

I(QR)

QR

Look at maxima of form factor

4

26

2

4

2

30

2

30

2

30

2

30

~1

~

4~1

4~

)cos()sin(4

)cos()sin(4~

qV

S

R

R

q

qR

qR

qR

qR

qR

qRqRqR

qR

qRqRqR

P

Surface of sphere


SoSe 2016, S. Roth

Porod´s Law

-1

0

1

2

3

4

5

6

-2.5 -2 -1.5 -1 -0.5 0

log q [nm-1]

log

I [

a.u

.]

SAXS

q -̂4

q -̂4

USAX

I(qR)

qR

R>1µm R~18nm

𝑃 𝑞𝑅 > 4.5 = 2𝜋𝑆

𝑉𝑃2𝑞−4

> Depends only on Surface and particle Volume

> No shape dependance


SoSe 2016, R. Roth

The structure factor – many particle, close distance

> Real systems: not dilute, many particles…

> Generalisation of Bragg‘s Law in crystallography:

I(q)= c P(q) S(q)

> Periodic ordering with periodicity d,x in the electron density :

> I(q) shows a corresponding maximum at q=2/(Dmax , x)

𝑆 𝑞 ∝ 1 − exp(−2𝜎 𝑞²𝐷

2 )

1 − 2 exp −𝜎 𝑞²𝐷2 cos 𝑞𝐷𝑚𝑎𝑥 + exp(−2𝜎𝐷

2𝑞2)

Form factor Structure factor

Interference due to assembly of particles

Lode (1998)

Roth et al., J. Appl. Cryst. 36, 684 (2003)

2R

Dmax,x

Smearing Distance of particles


SoSe 2016, R. Roth

The Structure factor – many particle, close distance

> Real systems: not dilute, many particles…

> Generalisation of Bragg‘s Law in crystallography:

I(q)= c P(q) S(q)

> Examples: R=5nm, Dmax=100nm, 25nm, sD/Dmax=25%

Low F P(q), S(q)=1 High F S(q)P(q)

q [nm-1] q [nm-1]


SoSe 2016, S. Roth

Structure factor and form factor

> Dmax=25nm

Dmax=10nm

> sD= 5nm, 1nm, 0.1nm

> 𝑆 𝑞 1𝑞 ∞

well separated particles


SoSe 2016, S. Roth

Colloidal System

> Latex spheres in water

I(q)= c P(q) S(q)

Low F P(q), S(q)=1 High F S(q)P(q)

q [nm-1]

> Gaussian distribution

of particle sizes

> Shift in maximum:

Decreasing distance

Hu et al., Macromolecules, 41, 5073 (2008)

Documents

31.05. : Small-Angle X-ray Scattering (SAXS) 02.06. : …photon-science.desy.de/sites/site_photonscience/content/e62/e... · T-SAXS vs GISAXS - Easy measurement - Easy analysis -