Small Angle Neutron Scattering:

First Steps…

Henrich Frielinghaus

Jülich Centre for Neutron Scattering

München - Garching

CC

CC

C CC C

C

C C

C

CC

CC

H

H

H

HH H

HH

HH

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

HH

CC

CC

C

C CC C

C

C C

C

CC

CC

H

H

H

HH H

HH

HH

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

HH

HH

HH

Scattering Length Density:

C

C

H H H HH

H

HHn

The selected Q-range will furthermore support this picture.

CC

O CC O

C

O C

C

CC

OC

H

H

H

H

H

H

H

H

H

H

H

H

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H

H

H

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H

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C CC C

C

C C

C

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CC

H

H

H

HH H

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H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

HH

HH

HH

Contrasts:C

C

CC

C

C CC C

C

C C

C

CC

CC

D

D

D

DD D

DD

DD

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

DD

DD

DD

OC

C

CC OHH

HH

H

H HH

H

HH

HHneutrons

Element SL [fm] Electron #

C 6.646 6

H -3.741 1

D 6.671 1

O 5.803 8

X-rays

SALS:refractiveindex

1st Born‘s Approximaiton:

)exp()()( 3iQrrrdQA ⋅= ∫ ρ

Scattering Amplitude ρ1

ρ2

Single Sphere

)(1

)(2

QAV

Qd

d

tot

=Ω

Σ

Scattering Intensity

Single Sphere(colloidal particle)

Loss of phase !

SANS geometry

λπ /210 == kk

4sin( /2)Q

πΘ

λ=

- Monochromator

- Position Sensitive Detector

Fourier Transformation:

φ 1

i

a

φ

Valuation of exp(…) Argument space (a,φ)

exp(iφ) = cos(φ) + i·sin(φ)

exp(a+iφ) = exp(a)·(cos(φ) + i·sin(φ))

Absolute value: exp(a)

We need just the oscillating

part for a=0 !!!

Fourier Transformation:

ρ(r)Often: symmetric

arrangement

r

substance: A B C D C B A

solvent

)cos()(

)exp()()(

Qrrdr

iQrrdrQA

⋅=

⋅=

∫

∫ρ

ρ

Fourier Transformation:

0.2

0.4

0.6

0.8

1.0

1.2

Scat

teri

ng le

ngth

den

sity

ρ(r

) [a

.u.]

0.4

0.6

0.8

1.0

1.2

Scat

teri

ng le

ngth

den

sity

ρ(r

) [a

.u.]

~a

~a

-5 -4 -3 -2 -1 0 1 2 3 4 50.0

0.2

Scat

teri

ng le

ngth

den

sity

Real space r-5 -4 -3 -2 -1 0 1 2 3 4 5

0.0

0.2

Scat

teri

ng le

ngth

den

sity

Real space r

Qa

Qaa

QrdrQA

a

a

)sin(2

)cos()(

⋅=

= ∫−

)exp(

)cos()/exp()(

22

22

Qaa

QrardrQA

−⋅⋅=

⋅−= ∫π

1x10-4

10-3

10-2

10-1

100

Sc

atte

ring

Int

ensi

ty |A

|2

Fourier Transformation:

Q-2

Planar surfaceO

OH4

OOH

4

OOH

4

OOH

4

OOH

4

OOH

4

OOH

4

OOH

4

OOH

4

surfactant film:

0.1 1 1010-6

1x10-5

Scat

teri

ng I

nten

sity

|A|

Q⋅a

diffuse surfaceO

OH4

OOH

4

OOH

4

OOH

4

OOH

4

OOH

4

OOH

4

OOH

4

OOH

4

More realistic: diffuse surface & center part: crossover from Q-2 to exp(-Q2)

Fourier Transformation of a sphere:

tiQrrdrdt

iQrrdrdd

rQidzdydxQA

)exp(2

))cos(exp()sin(

)exp()(

21

2

sphere

∫∫

∫ ∫∫

∫

⋅⋅=

=

=

π

ϑϑϑϕ

rr

Qr

rrϑ

ϕ

)cos(ϑ=t

Qr

Qrrdr

tiQrrdrdt

R )sin(22

)exp(2

0

2

1

∫

∫∫

⋅⋅=

⋅⋅=−

π

π )cos(ϑ=t

123123123123

Scattering of a shell4πr2: surface of a shell

)(rρ

general case

33

)()cos()sin(

33

4)(

QR

QRQRQRRQA

−⋅

=

π

ρ1

ρ2

1st Example: N independent spheres

)()(

)cos()sin(3

34

)()( 233

211 QQR

QRQRQRRQA δρ

πρρ +

−⋅

−=

Scattering Amplitude (1 sphere)

4444 34444 21)(

)()cos()sin(

)()()(2

331

21

21

QF

QR

QRQRQRVQA

V

NQ

d

d

tot

−⋅∆⋅==

Ω

Σρφ

Scattering Intensity (N spheres)

ρ1

ρ2

1st Example: independent spheres

2

331 )(

)cos()sin()(

−=

QR

QRQRQRQF

Formfactor

100

Reciprocal space:

0.1 1 1010-5

10-4

10-3

10-2

10-1

10

F(Q

R)

QR

Reciprocal space:

Qmin= 4.493 / R

For small Q:F(Q) ≈ 1 – (QR)2/5

ρ1

ρ2

1st Example: independent spheres

Polydispersity (important for soft matter)

100

0.1 1 1010-5

10-4

10-3

10-2

10-1

10

F(Q

R)

QR

Porod Q-4 law

general forcompact volumes

Some Examples for power laws:Q-1: cylinder, rods, (locally: worms) 1d (2π/L;2π/lP) < Q < 2π/d

Q-2: planes or spherical shells 2d 2π/L < Q < 2π/d

These examples are mass fractals with a maximum diameter L and a finite thickness d. The exponent can take any value between -1 & -3.

Q-2: Polymer (2d) 2π/U < Q < 2π/lP

Q-3: house of cards (3d) 2π/U < Q < 2π/L

These examples are fractals in 3d-space with an upper dimension U.

Q-4: Regular surface of a 3d-object 2d 2π/d < Q < 2π/dr

This example is 2d-surface fractal with an exponent 6–dim = 4.The limits are the smallest dimension d of the 3d-object and a further dimension dr where roughness would become visible.Then the exponent would be -4..-6, or gaussian diffuseness appears.

Summary

Many stages of structure(SAS smears atomic structure out)

Scattering angle Q is variable of SAS experiment(reciprocal space)

Fourier transformation of SLD ―› observed intensity|A|2 loss of information|A|2 loss of information

Droplet microemulsions

with CO2

Droplet microemulsions with CO2

Steel A. Ulbricht, Rossendorf

• Mixtures of Cr, Fe, Y2O3

• Milling (5h)• Sintering (spark plasma sintering)

ODS (Oxide Dispersion Strengthened) Steel• High Temperatures

not irradiated

irradiated

100

1000

10000 with magnetic field without magnetic field difference spheres

-1]

0.01 0.1

1E-3

0.01

0.1

1

10

100

dΣ/d

Ω [

cm-1

Q [Å-1]

Background

• Bovine fetuin (BF) is a glycoprotein that is

expressed in the liver.

• It is a systemic inhibitor of soft tissue

calcification.

Fetuin deficient mouse1 month: as healthy mouse 3 month: calcification started in blood

vessel6 month: calcification of blood vessel

and soft tissue11 month: total calcification

Structure of calciprotein particles (CPP)

Protein

Mineral

Water

Partial structure functions Pi,j(Q)

Second stage

-3

10-2

10-1

100

101

cm3 ]

P

self-terms

, , ,,

( ) ( )i k j k i j

i jk

dQ P Q

dρ ρ

Σ= ∆ ∆

Ω∑

, ( )M MP Qself-terms:

, ( )M PP Q10-3 10-2 10-1

10-4

10-3

10-2

10-1

100

10-4

10-3

Q [Å-1]

PM,P

cross-term

P ij(Q)

[10-2

0 cm PM,M

PP,P

,

,

( )

( )M M

P P

P Q

P Q

cross-terms:

From PM, P(Q) the position of the proteins with respect to the mineral can be

distinguished

Contrast variation

, ( ) 0M PP Q >

For second stage: