Methoden moderner Röntgenphysik II -...

Methoden modernerRöntgenphysik IIStreuung und Abbildung

Stephan V. RothDESY12.05.2015

Stephan V. Roth | Moderne Methoden der Röntgenphysik II | 12.05.2015 |

Outline

> 12.05. : Small-Angle X-ray Scattering (SAXS)

> 19.05. : Applications & A short excursion into Polymeric materials

> 21.05. : Grazing incidence SAXS (GISAXS)

> 02.06. : The polymer-metal interface – application of GISAXSOn the route to organic electronics

> 04.06.: In-situ studies of metal layer growth


T-SAXS vs. GISAXS

- Easy measurement- Easy analysis- In-plane information (qy,qz)- Any possible scattering from

substrate- Transparency of substrate- High energy

- 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

x

yz

qx

qyqz

2

Beamstop x

y

z

qx

qyqz

2i

f

2

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


> To understand the structure – property relation of materials onmultiple length scales

- Real pieces& materials

- Model systems- Nanotechnology

- q-resolution- Maximum q-value- Beam size

Aim

http://news.thomasnet.com/companystory/GE-Gas-Turbine-Technology-Selected-for-Pearl-GTL-Project-in-Qatar-495497

Courtesy: R. Gilles (TUM)


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


> Differential cross section

> Scattering occurs due to density differences

d

L

2I0

Detector → I

V=Sample volumeQ

ki

kf

2

if kk

Cross-section

2

Ω

Ω ⇨ΣΩ

1 σΩ

- 22

sin


d / resolution [Å]

1 10=1nm

100 1000 10000=1µm

100000=10µm

WAXS: Crystal

structure

WAXS, SAXS, GISAXS…Source: Streumethoden zur Untersuchung kondensierter Materie1996; ISBN 978-3-89336-180-9

WAXS

qxd=1

2 sin

1.54Å

SAXS SAXS/GISAXS:density fluctuations, precipitates

> R~particles “radius“

> d~interatomic distance

> SAXS: < 5°

Limit Guinier Porod Bragg

Log qd

R

q xR=1

Log Iq)


Scattering amplitude

( r ), V, VP

0

ir

rkieArA

0)(

ii ek ˆ2

fk

2dVri )(I0,

q

ki

kf kf 2

> Interference in far field

> Phase difference:

> Scattering amplitude:

> Intensity:

Δ ∙ ∙

3

= 2

4sin


)(rp

Single particle: Fourier transformation

Particle distribution function G(r)

i

ri

Electron density distribution

Scattering amplitudes of the whole arrangement

Scattered Intensity

Form factor Structure factor

Form factor and structure factor: Fourier transform

= 2

3

′ ′ 3 ′ ∗

∗ ] 3

∙


Two-phase model: Dilute systems

> Only form of particle relevant

> Matrix M, volume fraction Particles P , volume fraction (1-)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

>> R

2R

2R

ASAXS


Two phase Model

> Scattering amplitude:3 = 3 3

3

Δ 3

> = 2~∆

> Porod Invariant Q (Porod, 1982):Q 3 4 Φ 1 Φ Δ 2

> Only dependent on density contrast

Ableiten!Mittelung <..> erklären S.25, S.51


Herleitung Porod-Invariante

> Siehe Handzettel und Übung

> Q-Berechnung Übung


Two phase Model – single particle approximation

> Amplitude: Δ 3

> Intensity: = 2

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

> Incoherent sum of intensities:

~

Δ 21

3

nPfP=P

nMfM =M

r

2RnPfP

nMfM2R

Δ

VP~R3


Two phase Model – single particle approximation

> Amplitude: Δ 3

> Intensity: = 2

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

> Incoherent sum of intensities:

2

3)()cos()sin(3)(

qRqRqRqRqP

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

~

Δ 21

3


Colloid: homogeneous sphere of radius R

drddrerderqF rqiR

lumeparticleVoV

rqi

)sin()()( 2

0

2

0 00

3

R iqriqriqr

R

drrqr

eedrddreqF0

20

2)cos(

0 00 )sin(2)sin(2)(

RRR

drqqr

qqrr

qdrrqr

qqF

00

0

00

)cos()cos(4)sin(22)(

A simple, but important calculation:

30

32

0 )cos()sin(4)sin()cos(4)(qR

qRqRqRRqqR

qqRR

qqF


Colloid: homogeneous sphere of radius RPqR

qR

qR

PqR


Guinier radius

> Q→ 0

> Homogenous sphere of radius R

> Radius of gyration: replace homogenous sphere by shell of same moment of intertia: Rg

> ⁄

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

> Independent of particle form

3sin

3 ~115

2 2~exp15

2 2

~exp13

2 2

Ableiten


)3

exp()(lim2

222

0

g

q

RqVqI

Radius of Gyration Rg

Monodisperse spheres of radius R: RRg 5/3

qR

I(q)

2nm Colloidsdomains

Guinier Approximation

Roth et al., Appl. Phys. Lett. 91, 091915 (2007)


Porod‘s law: large q

Scattered intensity:

2

303 )cos()sin(4~

qRqRqRqRR Look at maxima of form factor

4

26

2

4

2

30

2

30

2

30

2

30

~1~

4~14~

)cos()sin(4

)cos()sin(4~

qVS

RR

q

qRqR

qRqR

qRqRqRqR

qRqRqRqR

P

Surface of sphere

I QR

QR


> Depends only on Surface and particle Volume

> No shape dependance

qR

I qR

-10123456

-2.5 -2 -1.5 -1 -0.5 0

log q [nm-1]

log

I [a.

u.]

SAXSq -̂4q -̂4USAX

R>1µm R~18nm

Porod‘s Law

4.5 2 2

qR


> Real systems: not dilute, many particles…

> Generalisation of Bragg‘s Law in crystallography:I q cP q S q

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

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

Form factor Structure factor

Interference due to assembly of particles Dmax,

2R

The structure factor – many particles, close distance

Distance of particlesSmearing

Lode (1998)Roth et al., J. Appl. Cryst. 36, 684 (2003)


> Real systems: not dilute, many particles…

> Generalisation of Bragg‘s Law in crystallography:I q cP q S q

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

The structure factor – many particles, close distance

Low P q ,S q 1 High S q P q

q nm‐1 q nm‐1

Iq


Structure factor and form factor

> Dmax=25nmDmax=10nm

> D= 5nm, 1nm, 0.1nm

> →1 →∞well separated particles


Colloidal systems

q [nm-1]

> Gaussian distributionof particle sizes

> Shift in maximum:Decreasing distance

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

Low P q ,S q 1High S q P q

> Latex spheres in waterI q cP q S q

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