Small-Angle Neutron Scattering - Kfki

Small-Angle Neutron Scattering 

Mitglied in der Helmholtz-Gemeinschaft 

1 

Jülich Centre for Neutron Science 

Forschungszentrum Jülich GmbH 

2 

Frank Laboratory of Neutron Physics 

Joint Institute for Nuclear Research 

29. May 2013 | Artem Feoktystov 1 , Mikhail Avdeev 2


Plan 

1. Small-angle neutron scattering 

2. Contrast variation. Basic functions 

3. Theory 

4. Core-shell structure 

5. Magnetic fluids 

6. SANS of polarized neutrons 

7. Spin canting 

8. Direct modeling 

9. Conclusions

Small-angle neutron scattering (SANS) 

Non-polarized neutrons 

= 0 

k 

qsinφ 

qcosφ 

k r k r 

0 

θ 

q r 

r 

q = k r 

−k r 

0 

4π 

θ 

q= 

sin 

λ 2 

θ < 

10 

o 

k 0 

For diluted samples (ϕ m < 3 vol. %) 


H r ) 

Differential 

cross section 

dσ 

d 

Ω Ω 

= 

dσ 

(q 

dΩ 

dσ 

( q) 

dΩ 

2 

≈ FN ( q) 

+ 

Nuclear formfactor 

2 

3 

F 

2 

M 

( q) 

Magnetic formfactor

SAS: obtained information 

Scattering length density (SLD) 

profile 

Particle structure 

(form-factor F(q)) 

I(q)~F 2 (q)S(q) 

Scattering 

curve 

q 

Radial distribution function (RDF) 


Particle interaction 

(structure-factor S(q))

Small-Angle Neutron Scattering (SANS): 

modern research objects 

Biological macromolecular complexes 

Self-organization in solutions of 

surfactants and lipids, complex liquids 


Liquid dispersions of organic and 

magnetic materials 

Polymers 

Tendency – study of complex mixed systems! 

Features of neutron scattering: 

•wide abilities of isotopic hydrogendeuterium 

substitution 

•magnetic scattering 

•high penetration depth (absence of 

special sample preparation)

Neutron and X-ray scattering lengths 

Mitglied in der Helmholtz-Gemeinschaft

Contrast variation: basic idea 

“Core-Shell” particles 

in “Solvent” 


Matching of scattering 

from “Shell” 

Matching of scattering 

from “Core” 

In SANS the realization is made by substitution of H with D in solvent!

Monodisperse non-magnetic systems: 

basic functions approach in contrast variation 

Contrast 

∆ρ 

= 

ρ − 

ρ s 

H.B. Stuhrmann (1975) 

average scattering length 

density of the particle 

scattering length 

density of the solvent 


shape scattering 

scattering from density fluctuation 

r r 

ρ ( ) = ρ( 

) − ρ 

f

Example: core-shell particles 

ρ 1 

, R 1 

ρ 

I ( q) 

= n[( 

ρ ρs ρ ρ Φ qR 

2 

1 

− ) V1Φ( 

qR1 

) − ( 

1 

− 

0) 

V0 

( 

0)] 

Φ( x) 

= 3(sin x − x cos x) / x 

n is the particle number density 

3 

V 

i 

= 

3 

( 4 / 3) πRi 

ρ 0 

, R 0 

ρ s 

ρ = 

V 

V 

V 

( ρ 

0 0 

ρ0 

+ 1− 

) 

1 

V1 

1 

∆ρ 

= 

ρ − 

ρ s 


I( q) = nV ( ρ − ρ ) [ Φ( qR ) − Φ ( qR )] + 

2 2 2 

0 1 0 1 0 

+ 2 nVV ( ρ − ρ ) Φ( qR )[ Φ( qR ) − Φ( qR )]( ∆ ρ) 

+ 

1 0 1 0 1 1 0 

+ nV Φ ( qR )( ∆ρ) 

2 2 2 

1 1 

Is 

Ics 

( q) 

( q) 

I ( ) c 

q

Core-shell particles. Basic functions 

R 0 = 4.5 nm 

R 1 = 6.0 nm 

ρ 0 = 7.0×10 10 cm -2 

ρ 1 = 0 cm -2 

ρ s1 = 5.4×10 10 cm -2 

ρ s2 = 3.0×10 10 cm -2 

ρ s3 = 1.0×10 10 cm -2 

5 

I 1 

(q) 

100 

10 

I c 

(q) 

4 

1 

I 3 

(q) 

0.1 


I(q), arb. units 

3 

2 

1 

0 

I 2 

(q) 

0.01 

1E-3 

0.01 0.1 1 10 

q, nm -1 1E-3 0.01 0.1 

0 

I 

I s 

(q) 

cs 

(q) 

0.01 0.1 1 

q, nm -1

Guinier approximation 

Guinier approximation, 1939 

ellipsoid 

sphere 

cylinder 

I( q ~ 0) = I( 0) 

e − 

q 

< 1 

R 

g 

2 2 

q R g 

3 


R g 

( ) 

2 

I( 0) = nV ρ − ρs 

2 

– scattering intensity in zero angle 

– radius of gyration of scattering density distribution in particle

Monodisperse systems: Guinier regime 

1 2 2 

I( q ~ 0) = I( 0)exp( − q R g 

) 

Forward scattering 

intensity 

q 

< 1 

3 Rg 

Radius of gyration 

2 2 

I( 0 ) = nV ( ∆ρ) 

R = R + α ∆ρ − β ( ∆ρ) 

c 

2 2 2 

g c 

I(0) 

R 2 g 

radius of gyration 

of the shape 

2 

R c 


0 

match point 

∆ρ 

0 

(∆ρ) −1

Radius of gyration: 

monodisperse systems 

R = R + α ∆ρ − β ( ∆ρ) 

2 2 2 

g c 

α = V 

−1 

c 

∫ 

V c 

r 

ρ ( r ) 

f 

r 2 

r 

d 

β = 

∫ ∫ 

−2 

Vc 

f 

r1 

) 

f 

( r2 

)( r1 

r2 

) dr1 

dr2 

V V 

c 

c 

ρ 

( ρ 

β = 0 

β > 0 

α > 0 

α < 0 


“light” 

component 

“heavy” 

component


Example: ribosome 

70S ribosome 

1 2 

50S 

subunit 

30S 

subunit 

Proteins + RNA

Contrast variation on ribosome 

30S subunit 

is deuterated 

both 50S and 30S 

subunits are 

deuterated 


50S ribosome 

H.Stuhrmann, et al., PNAS 73 (1976) 2379 

70S ribosome 

M.Koch, et al., Biophys. Struct. Mechan. 

4 (1978) 251

Systems with high polydispersity 


• Avdeev, M.V © 

What information one can obtain in general case from the SANS 

contrast variation?

Polydisperse non-magnetic systems 

I( 

q) 

= ∑ ni{ 

I 

i 

i 

s 

( q) 

i 

+ ∆ ρI 

( q) 

+ 

i 

cs 

( ∆ 

i 

ρ) 

2 

I 

i 

c 

( q)} 

Size polydispersity 

Structural polydispersity 

ρ 1 

ρ 2 

ρ 1 

ρ 2 


ρ 3 

ρ s 

ρ 3 

ρ s

Polydisperse magnetic systems 

I( 

q) 

i 

i 

2 i 

= ∑ ni { I 

s 

( q) 

+ ∆iρIcs 

( q) 

+ ( ∆iρ) 

Ic 

( q)} 

+ 

i 

I 

m 

( q) 

Size polydispersity 


ρ 1 

ρ 2 

ρ 1 

ρ 2 


ρ 3 

ρ s 

ρ 3 

ρ s

Polydisperse systems. General case 

I(q) = + + 

is averaging over the particle polydispersity function. 

The idea of the following transformations is to introduce the effective mean 

scattering length density, ρ e , independent of the averaging, so that the 

intensity equation takes the classical form but with the modified contrast. 

∆ ~ ρ = 

ρ e 

− ρ s 


∆ ρ ρ − ρ + ρ − ρ = ρ − ρ + ∆ 

~ ρ 

I( 

q) 

= 

= 

e e s 

e 

~ 

I 

s 

( q) 

+ ∆ 

~~ 

ρI 

cs 

( q) 

+ 

( ∆ 

~ ρ) 

2 

~ 

I 

c 

( q)

Modified basic functions 

I 

~ ( q) 

2 

s 

s 

e cs 

e 

~ 

I 

~ 

I 

cs 

c 

( q) 

( q) 

=

> + < ( ρ − ρ ) I ( q) 

> + < ( ρ − ρ ) I ( q) 

=

> + 2 < ( ρ − ρ ) I ( q) 

cs 

=

c 

> 

e 

c 

> 

c 

> 

Choice of 

ρe 


ρ 

e 

=< 

∂I 

(0) 

∂ρ 

s 

= 0 

effective match point 

2 

ρI 

c 

( 0) > / < I 

c 

(0) >=< ρVc 

> / < Vc 

2 

>

Modified shape basic function 

ρ s 

From three measurements with different : 

~ 

I ( q) 

c 

= −{( 

ρs2 − ρs3) 

I1( 

q) 

+ ( ρs3 

− ρs 

1) 

I2( 

q) 

+ ( ρs 

1 

− ρs2) 

I3( 

q)}/ 

/( ρ 

s1 − ρ 

s2 

)( ρ 

s2 

− ρ 

s3)( 

ρ 

s3 

− ρ 

s1) 

ρ e 

Knowledge of exact -value is not necessary 

to obtain shape scattering function! 


~ 

I (0) = n < 

~ 2 2 2 

R =< V R > 

c 

2 

c 

V c 

c 

c 

/ 

> 

< V 

2 

c 

>

Guinier invariants 

I (0) = n∆ % ρ < V > + 

2 2 

c 

2 2 

( ρ ρe ) 

c m 

(0) 

+ n < − V > + I 

~ 

R 

2 

g 

= 

2 2 

< Vc 

Rc 

> 

( 

2 

< V > 

c 

+ 

A 

~ − 

∆ρ 

B 

( ∆ 

~ ρ) 

2 

) /(1 + 

D 

( ∆ 

~ ρ) 

2 

) 

I(0) 

classical 


(a) 

~ 2 

R g 

< V 

< V 

R 

2 2 

c c 

2 

c 

> 

> 


0 

effective 

match point 

∆ρ ~ 

In comparison with monodisperse case there are possibilities 

for analyzing Guinier region around effective match point! 

− 

B / 

D 

0 

( ∆ρ~ 

) 

−1

Polydisperse systems 

Intensity at zero angle 

~ 

I 

~ 

I 

~ 

s 

I cs 

(0) 

(0) 

(0) 

= 

= 

= 

n 

n < ( ρ − ρ ) 

0 

< 

2 

c 

V c 

> 

e 

2 

V 

2 

c 

> 

I(0) 

classical 



I (0) 

= 

+ 

2 

n∆ρ~ 

< V 

2 

c 

n < ( ρ − ρ ) 

e 

> + 

2 

V 

2 

c 

> 

Full matching is not possible! 

0 

effective 

match point 

∆ρ ~

Polydisperse systems 

Radius of gyration 

~ 

R 

2 

g 

= 

2 2 

< Vc 

Rc 

> 

( 

2 

< V > 

c 

+ 

A 

~ − 

∆ρ 

B 

( ∆ 

~ ρ) 

2 

) /(1 + 

D 

( ∆ 

~ ρ) 

2 

) 

classical 


A 

= 1 2 

c 

< V 

2 

2 

( < αV 

> + 2 < ( − ) > )(a) 

2 c 

ρ ρ 

e 

Vc 

R 

> 

c 

1 

B = 

< V 

2 

c 

( < βV 

> 

2 

c 

2 

− < ( ρ − ρ ) V 

> − < ( ρ − ρ ) αV 

e 

2 

c 

R 

2 

c 

e 

> ) 

2 

c 

> 

~ 2 

R g 

< V 

< V 

R 

2 2 

c c 

2 

c 

> 

> 


D 

< ( ρ − ρ ) 

e 

= 

2 

< Vc 

2 

> 

V 

2 

c 

> 

In comparison with monodisperse case there are possibilities 

for analyzing Guinier region around effective match point! 

− 

B / 

D 

0 

( ∆ρ~ 

) 

−1

Case of two kinds of particles 

ρ ,ε ,ε 

1 1 

2 2 

ε + ε 

1 2 

= 

1 

2 

2 2 

ρ = ( ε ρ V + ε ρ V ) /( ε V + ε V 

e 

2 

1 1 c1 

2 2 c2 

1 c1 

2 c2 

) 

I(0) 

= 

ρ ~ 2 2 2 

2 2 2 2 2 

n∆ρ ( ε V + ε V ) + n( 

ρ − ρ ) ε ε V V /( ε V + ε V ) 

1 

c1 

2 

c2 

1 

2 

1 

2 

c1 

c2 

1 

c1 

2 

c2 


Residual intensity at zero angle is determined by the difference in SLDs.


V c 

= const 

=< ρ > 

I 

ρ e 

2 2 

2 2 ~ 2 2 2 2 

( 0) = n∆ 

~ ρ Vc 

+ n < ( ρ − ρe 

) > Vc 

= n∆ρ 

Vc 

+ nσ 

ρVc 

~ 2 2 A B 

Rg = ( Rc 

+ ~ − ~ ) /(1 + 

2 

∆ρ 

( ∆ρ) 

D 

( ∆ 

~ ρ) 

2 

) 

A =< α > 

B 

=< 

β > − < 

2 2 

2 2 

( ρ − ρe 

) α > − < ( ρ − ρe 

) > Rc 

=< β > − < ( ρ − ρe 

) α > −σ 

ρ 

Rc 


D =< ρ ρe 

>= σ 

2 2 

( − ) 

ρ

Example: ferritin 

protein shell, R 1 = 6 nm, 

ρ 1 = 1.9-2.95 × 10 10 cm -2 

ρ s 


ferrite core, R 0 = 4 nm 

SANS curves from 

iron unsaturated ferritin 

in various H 2 O/D 2 O mixtures 

H.Stuhrmann, E.Duee, JAC 8 (1975) 538


Density of the core is distributed with 

σ ρ = 0.35 × 10 10 cm −2 = −1.14 × 10 −3

For particle of two components with common center of mass: 

2 

2 ( ρ0 

− ρs 

)( V0 

/ V1 

) R0 

+ ( ρ1 

− ρs 

)(1 − ( V0 

/ V1 

)) R 

Rg 

= 

∆ρ 

2 

1 

I.N.Serdyuk, B.A.Fedorov, 

J. Polym. Sci. 11 (1973) 645 

2 2 

2 2 ( V0 

/ V1 

)(1 − ( V0 

/ V1 

))( ρ1 

− ρ0)( 

R1 

− R0 

) 

Rg 

= Rc 

+ 

∆ρ 

= 

R 

2 

c 

+ 

α 

∆ρ 

For iron unsaturated ferritin: 

~ 

R 

= ( R 

< α > 

∆ 

~ ρ 

B 

( ∆ 

~ ρ) 

) /(1 

2 2 

g c 

+ − + 

2 

D 

( ∆ 

~ ρ) 

2 

) 


2 

< α 

>= 

( V0 / V1 

)(1 − ( V0 

/ V1 

))( ρ1− < ρ0 

> )( R1 

− R 

< ρ 

>= 

e 

( V 

ρ 

0 

/ V1 

) < ρ0 

> + (1 − ( V0 

/ V1 

)) 

Analysis of B, D 

1 

2 

0 

) 

< ρ 

< ρ 

10 -2 

0 >= 

5 .0× 

10 cm 

10 -2 

0 >= 

4 .9× 

10 cm 

additional data for σ ρ

Magnetic neutron scattering 

Cross-section of the magnetic moment 

of one atom in domain: 

dσ 

dΩ 

2 2 

( ) = m 

vm 

p 

p - magnetic scattering length 

vm 

= γ −η ( ηγ ) 

- magnetic interaction vector 

Nuclear + magnetic scattering: 

dσ 

= 

dΩ 

b 

2 

+ 

2 2 

2bp( 

vm t ) + vm 

p 


unit vector along the 

magnetic moment 

direction 

ν m 

=sinα, α - angle between 

unit vector along 

the scattering vector 

γ 

and 

q 

neutron spin 

non-polarized beam: 

dσ 

= 

dΩ 

b 

disoriented domains: 

dσ 

dΩ 

2 

+ v 

2 

m 

p 

b 

2 v 

2 p 

2 b 

2 

= + < 

m 

> 

Ω 

= + 

2 

2 p 

3 

2

Effect of magnetic scattering 

Monodisperse case 

∆ 

~ ρ = ∆ρ 

~ 

Ic ( q) 

= Ic( 

q) 

~ 

I 

s 

( q) 

= Is 

( q) 

+ I 

~ 

I ( q) 

I ( q) 

cs 

= 

cs 

m 

( q) 

I 

I 

= 

2 

3 

ρ 

2 2 

m 

( 0) nV m m 

m 

( q 

~ 0) = 

I 

m 

(0)(1 − 

R 

2 q 2 

m 

/ 3) 

~ 

I 

(0) 

= 

2 

3 

2 

s 

nV m 

ρ 

2 

I 

s 

( q ~ 0) = I 

m 

(0) −{ 

−βnVc 

+ 

2 

m 

~ 2 

I (0) R } q 

2 

m m 

/ 3 


~ 

R 

2 

g 

= 

( R 

2 

c 

+ 

α 

− 

∆ρ 

I 

∆ρ + 

2 

3 

ρ 

2 2 

2 2 

( 0) = n Vc 

n 

mVm 

2 

β − (2 / 3) ρ 

m 

( V 

( ∆ρ) 

2 

m 

2 

/ V 

2 

c 

) R 

2 

m 

) /(1 + 

2 

(2 / 3) ρ 

m 

( V 

( ∆ρ) 

2 

m 

2 

/ V 

2 

c 

) 

)


Monodisperse case. 

Radius of gyration: homogeneous particles 

α = 0 β = 0 

R 2 g 

2 2 2 2 

~ 2 2 (2 / 3) ρm( 

Vm 

/ Vc 

) Rm 

R 

g 

= ( Rc 

+ 

) /(1 + 

2 

( ∆ρ) 

a 

2 

(2 / 3) ρm( 

V 

( ∆ρ) 

2 

m 

2 

/ V 

2 

c 

) 

) 

The view depends on which radius is larger! 

R 2 m 

R m 

> R c 

If for spherical particles 


R 2 c 

R 2 m 

0 

R m 

< R c 

(∆ρ) -1 

R 

m 

= R c 

~ 2 2 

g 

R c 

R = 

No magnetic scattering effect!


Polydisperse case 

Size-polydisperse non-homogeneous particles 

2 2 

2 2 

2 

I ( 0) = n∆ρ~ 

< Vc 

> + n < ( ρ − ρe) 

Vc 

> + (2 / 3) nρm 

< Vm 

2 

> 

~ 

R 

2 

g 

= 

2 2 

< Vc 

Rc 

> 

( 

2 

< V > 

c 

+ 

A 

~ − 

∆ρ 

B 

( ∆ 

~ ρ) 

2 

) /(1 + 

D 

( ∆ 

~ ρ) 

2 

) 


B 

1 2 

2 

A = ( < αV 

> + 2 < ( − ) 

2 > ) 

2 c 

ρ ρ 

e 

Vc 

R 

< V > 

c 

c 

= 1 2 

2 

2 2 2 

2 2 

( < βV 

> − < ( − ) > − < ( − ) > −(2 / 3) < 

2 > 

2 c 

ρ ρe 

αVc 

ρ ρe 

Vc 

Rc 

ρm 

Vm 

R 

< V > 

m 

c 

2 2 

2 2 

< ( ρ − ρe) 

Vc 

> (2 / 3) ρm 

< Vm 

> 

D = + 

2 

2 

< Vc 

> < Vc 

> 

)

Basic functions approach 

ρ = 

const 

bi 

ρ = ∑V 

i 

ρ ≠ 

const 

ρ 

e 

2 

< ρV 

> 

= < 

2 

V > 

Contrast 

Modified contrast 

ρs 

∆ ρ = ρ − ρ s 

ρs 

∆ % ρ = ρ − ρ 

e 

s 

I( q) = I ( q) + ∆ ρI ( q) + ( ∆ρ) 2 I ( q) 

I( q) = I % ( q) + ∆ % ρI % ( q) + ( ∆ % ρ) 2 I % ( q) 

s cs c 

s cs c 

∆ ρ = ∞ ∆ % ρ = ∞ 


I ( ) c 

q 

• Stuhrmann, H.B.(1982). Small-Angle X-ray Scattering, edited by O. Glatter & O. Kratky, pp.197-213, London: Academic Press 

• Avdeev, M.V. J. Appl. Cryst. 40 (2007) 56-70 

I% 

( q) =

> 

c 

c

Integral scattering parameters 

2 2 

2 2 2 2 

I( 0 ) = nV ( ∆ρ) 

I( 0) = n < V > ( ∆ % ρ) + n < ( ρ − ρ ) V > + 

c 

Scattering intensity in zero angle: 

c e c 

2 2 

+ ( 2 3) 

nρ 

< V > 

m 

m 

I(0) 

I(0) 


0 ∆ρ 

0 ∆ρ

Integral scattering parameters 

R 

α 

= R + − 

∆ρ 

2 2 

g c 

Radius of gyration: 

2 2 

β 

2 

⎛ < Vc 

Rc 

> A B ⎞ ⎛ D ⎞ 

R 

2 g 

= ⎜ + − ⎟ + 

( ∆ρ) 

Vc 

ρ ( ρ) ⎜1 

2 2 2 

< > ∆ ∆ ( ∆ρ) 

⎟ 

⎝ 

% % ⎠ ⎝ % ⎠ 

α0 

R 2 g 

R 2 g A

Magnetic fluids (ferrofluids) 


Magnetic fluid: biomedical applications 

The main direction is cancer treatment! 

Drug delivery 

Therapy 


Diagnostics

Magnetic drug concentrator: 

idea (SIEMENS) 


Magnetic drug concentrator: 

prototype (SIEMENS) 


Magnetic fluids (ferrofluids): 

specific properties for biomedicine 

Magnetic nanoparticles, size d = 2-30 nm 

(one-domain magnetic state → 

magnetic moment of single particle ~10 3 -10 5 µ B 

→ superparamagnetic behavior) 

Liquid carrier 

Surfactant shell, 

thickness d = 1-2 nm 


R.E. Rosensweig, 1966 

Small size – large specific surface, 

natural removal from organisms 

Combination of liquidity 

and magnetism are widely 

used in technical applications 

Magnetism – control by external 

magnetic field

Classical stabilization procedure for 

nonpolar organic magnetic fluids 

Co-precipitation 

aqueous solutions Fe 2+ , Fe 3+ with NH 4 OH (T = 80 o C) 

Oleic Acid (OA) 


Coating 

chemisorption of oleic acid (T = 80 o C) 

Nanomagnetite 

Washing 

water 


Flocculation 

acetone 

Initial dispersing 

light hydrocarbon 

Repeating flocculation and re-dispersing 

surfactant removal and particle concentrating 

Organic liquid 

(hydrocarbon) 

Organic non-polar magnetic fluids are 

often the first step in synthesis of 

various biocompatible systems

Types of ferrofluid stabilization 

Single sterical 

stabilization 

Double sterical 

stabilization 

Ionic (electrostatic) 

stabilization 

H + 

H + 

H + 

H + 

H + H + H + H + 

H 

H + + 

H + H + 


Surfactant chemisorption; 

no surfactant excess 

I layer – surfactant chemisorption; 

II layer – physical adsorption of 

surfactant in excess 

Adsorption of ions 

(H + , ОH − , citrate-ion) 

(Massart’s method)

Surfactants for stabilization 

of magnetic fluids based on organic liquids 

Unsaturated monocarboxylic acid 

Kinked 

double bond 


СН 3 (CH 2 ) 8 =(СН 2 ) 8 СOОН or C 18 H 34 O 2 

Excellent stabilizer 

Saturated monocarboxylic acids 


Stearic Acid (SA) 

СН 3 (CH 2 ) 16 СOОН or C 18 H 36 O 2 

Extremely bad stabilizer 

Palmitic Acid (PA) 

C 16 H 32 O 2 

Myristic Acid (MA) 

C 14 H 28 O 2 

Lauric Acid (LA) 

C 12 H 24 O 2 

Intermediate 

case

Nuclear and magnetic particle form-factors 

Nuclear scattering 

Magnetic scattering 

F 

2 

( q) 

= [ 

R 

max 

∫ 

0 

sin( qr) 

( ρ( 

r) 

− ρs 

) 

qr 

4πr 

2 

dr] 

2 

F 

2 

N 

( q) 

= [( ρ0 

- ρ1) 

V0Φ( 

qR0 

) + ( ρ1 

- ρs 

) V1Φ( 

qR1 

)] 

2 

F 

2 

M 

( q) 

= [ ρmVmΦ( 

qRm 

)] 

2 


Model spherical “core-shell” 

Model “homogeneous sphere” 

Unmagnetized FF 

Φ( x) 

= 3(sin( x) 

− x cos( x)) / x 

I + 

2 

3 

2 

2 

( q) 

= nF N 

nFM 

3

Contrast variation on MF: principle stages 

Dissolution of initial concentrated system 

(ϕ m 1-10%) with mixtures of H- and D- carriers; 

measurements of scattering curves. 

Determination of 

effective match point 

from minimum of dependence I(0) ~ ρ s 

Minimization of 

χ 

2 

Experimental separation 

on basic functions 

1 [ I ( q) I ( q) I ( q) ( ) I ( q)] 

= ∑ % % % % % 

N − 3 ( ) 

k 

2 2 

k 

− 

s 

− ∆ρk cs 

− ∆ρk c 

2 

σ 

k 

q 

I k 

(q), σ k 

(q) are experimental data for k-th contrast; 

N is number of contrasts. 

SLD map for MF components 


Separation to modified basic functions 

Comparison of I s with scattering 

in effective match point 

(check-out separation) 

Analysis of I c . 

Comparison of I c (total shape) 

with scattering in H-carrier 

(only «sphere» of magnetic material) 

7.0×10 10 cm -2 

2.0×10 10 cm -2 D-solvent 

1.0×10 10 cm -2 

0.0×10 10 cm -2 

magnetic SLD 

Н-solvent 

magnetic 

materials 

surfactants

Contrast variation on FFs: 

separate nonhomogeneous particles 

Intensity, cm -1 

10 

1 

0.1 

0.01 

10 

Magnetite (ϕ m ~ 1 %) coated with oleic (OA) or myristic (MA) acids in benzene 

OA 

10% C 6 

D 6 

20% C 6 

D 6 

30% C 6 

D 6 

50% C 6 

D 6 

55% C 6 

D 6 

60% C 6 

D 6 

70% C 6 

D 6 

75% C 6 

D 6 

90% C 6 

D 6 

q, nm -1 

0.1 1 

100% C 6 

D 6 

100% C 6 

D 6 

90% C 6 

D 6 

75% C 6 

D 6 

60% C 6 

D 6 

I (0) 

2 

R g 

I(0) for OA, cm -1 

I(0) для ОК, см -1 

35 

30 

25 

20 

15 

10 

5 

0 

80 

70 

60 

МК 

MA 

34.2 % 

ОК 

OA 

63 % 

0 

0.0 0.2 0.4 0.6 0.8 1.0 1.2 

ОК 

OA 

МК 

MA 

[C 6 

D 6 

]/[C 6 

D 6 

+C 6 

H 6 

] 

10 

8 

6 

4 

2 

I(0) для for MA, МК, см cm -1 -1 



1 

0.1 

0.01 

MA 

0.1 1 

q, nm -1 

50% C 6 

D 6 

40% C 6 

D 6 

38% C 6 

D 6 

35% C 6 

D 6 

32% C 6 

D 6 

30% C 6 

D 6 

20% C 6 

D 6 

10% C 6 

D 6 

R 

2 

R 2 g , 

g 

nm 

2 

,нм2 

50 

40 

30 

20 

10 

0 

-10 

-18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 

(∆ρ) -1 −1 ×10 -10 ,,cm см 22 

/


separate nonhomogeneous particles 

Separation on modified basic functions 

Comparison of I s (q) with the scattering 

at the effective match point 

1.1 

1.0 

0.9 

0.8 

0.7 

Ic 

1 

MA 

MA 35% C 6 

D 6 

OA 

OA 60% C 6 

D 6 

0.6 

0.5 

0.4 

0.3 

0.2 

Ics 

Is 

I s 

, cm -1 

0.1 

0.1 

0.0 


-0.1 

0.1 1 

q, nm -1 

0.1 1 

q, nm -1


separate non-homogeneous particles 

Magnetite (ϕ m ~ 1 %) coated with oleic (OA) or myristic (MA) acids in benzene 


I I c 

x см c × 10 -20 , cm 3 

Analysis of I с (q) and comparison with 

scattering from magnetite component 

10 

1 

0.1 

0.01 

1E-3 

1E-4 

D N 

(R) 

0.4 

0.3 

0.2 

0.1 

from Н-solvent 

MA 

МК 

OA 

ОК 

0.0 

0 2 4 6 8 10 

R, нм 

ОК 

МК 

from I c (q) 

q, нм nm -1 -1 

0.1 1 

OA 

MA 

I c (q) 

I(q) 

at 0% C 6 D 6 

Size and polydispersity decrease when substituting OA with MA! 

Thickness of effective surfactant shell (1.5 nm) does not change with substitution! 

• Feoktystov, A.V., Avdeev, M.V., Aksenov, V.L., Bulavin, L.A., et al., Surface Investigations 3 (2009) 1-4 

• Feoktystov, A.V., Avdeev, M.V., Aksenov, V.L., Bulavin, L.A., et al., Solid State Phenomena 152-153 (2009) 186-189

SANS on water-based ferrofluids 


100 

10 

1 

0.1 

magnetite/lauric acid/water 

0% D 2 

O 

10% D 2 

O 

20% D 2 

O 

30% D 2 

O 

40% D 2 

O 

50% D 2 

O 

60% D 2 

O 

70% D 2 

O 

80% D 2 

O 

90% D 2 

O 


10 

1 

0.1 

magnetite/myristic acid/water 

0% D 2 

O 

10% D 2 

O 

20% D 2 

O 

30% D 2 

O 

40% D 2 

O 

50% D 2 

O 

60% D 2 

O 

70% D 2 

O 

80% D 2 

O 

90% D 2 

O 

0.01 

0.1 1 

q, nm -1 

0.1 1 

q, nm -1 


I( q ~ 0) = I( 0) 

e − 

2 2 

q R g 

3 

Due to a strong aggregation in both samples 

the Guinier region is inaccessible! 

• Feoktystov, A.V., Bulavin, L.A., Avdeev, M.V., et al., Ukr. J. Phys. 54 (2009) 266-273

Dependence of match point on q 

Match Point, D 2 

O vol. % 

60 

50 

40 

30 

20 

10 

LA 

MA 

0 

0.0 0.2 0.4 0.6 0.8 1.0 

q, nm -1 


In case of myristic acid stabilization homogeneous aggregates with the size 

bigger than 20 nm are present in the system. 

In magnetic fluid with lauric acid stabilization the aggregates in the system 

are inhomogeneous.

Structure parameters of aggregates 


I c 

, x10 -20 cm 3 

10 

1 

0.1 

0.01 

1E-3 

0.1 1 

q, nm -1 

LA 

MA 

p(r) 

0.15 

0.10 

0.05 

Characteristic radii of aggregates: 

(19.5 ± 0.3) nm (LA+LA stabilization) 

(13.8 ± 0.1) nm (MA+MA stabilization) 

0.25 TEM of separate particles 

0.20 

Indirect Fourier Transform of I c 

(q) 

MA 

Indirect Fourier Transform of scattering 

curve of the sample on H 2 

O 

LA 

MA 

LA 

0.00 

0 10 20 30 40 50 

r, nm 

In case of lauric acid stabilization nanomagnetite is coated with 

a surfactant shell with thickness ~3.5 nm. 

In the sample with myristic acid stabilization the aggregates in the system 

consist of nanomagnetite with one incomplete layer of the acid.


Aggregate structure of both ferrofluids 

magnetite/lauric acid/water 

magnetite/myristic acid/water


separate homogeneous particles 

Biocompatible FFs with maghemite coated by citric ions (0.2 nm) in water. 

Coulomb interaction compensated (NaCl addition). Magnetic interaction compensated (ϕ m ~ 1%). 

I c (q) 

(negligible magnetic scattering) 


I(q), cm -1 

100 

10 

1 

0.1 

0.01 

1E-3 

1E-4 

0 % D2O 

70 % D2O 

80 % D2O 

90 % D2O 

95 % D2O 

100 % D2O 

I s (q) 

(matched nuclear scattering) 

0.1 1 

q, nm -1 

R g 

2 

110 

100 

90 

80 

70 

60 

50 

characteristic nuclear radius 

R N ~ 10 nm 

0 2 4 6 8 10 

∆ρ~ 

−1 

( ) × 

Increase in apparent particle size 

because of residual Van der Waals interaction 

(non-uniform surface charge distribution) 

• Avdeev, M.V., Dubois, E., Mériguet, G., et al., J. Appl. Cryst. 42 (2009) 1009-1019 

characteristic magnetic radius 

R M ~ 7.2 nm 

10 

-10 

cm 

2

Small-angle neutron scattering 

Polarized neutrons – magnetic structure 

qsinφ 

H r ∞ 

k 

qcosφ 

k r q r 

θ 

k r 0 

r 

q = k r 

−k r 

0 

4π 

θ 

q= 

sin2 

λ 

k 0 


Incident beam of 

polarized neutrons with 

wavelength λ 

d 

dΩ 

− 

d 

dΩ 

Dissolved samples. 

Cross section for two states of neutron polarization 

+ 

σ r 2 

2 

2 

( q) 

≈ FN 

( q) 

+ { FM 

( q) 

− 2FN 

( q) 

FM 

( q)}sin 

σ r 2 

2 

2 

( q) 

≈ FN ( q) 

+ { FM 

( q) 

+ 2FN 

( q) 

FM 

( q)}sin 

Nuclear formfactor 

Magnetic formfactor 

ϕ 

ϕ

Small-angle neutron scattering: 

polarized neutrons 

Magnetite (ϕ m ~ 1%) with OA and MA coating in D-cyclohexane 

I + ,I − , H=0 I + , H=H ∞ I − , H=H ∞ 

OA 

2D pattern of neutron scattering for two states of 

neutron polarization: parallel (I − ) and antiparallel 

(I + ) to external magnetic field (2.5 Т) on 

samples: 

MA 

Detector size 55×55 cm 2 

Sample-detector distance 4.5 m 

Neutron wavelength 0.83 nm 


H r ∞

Separation of nuclear and 

magnetic neutron scattering 

D-cyclohexane + Fe 3 

O 4 

/ OA (ϕ m ~ 1%) 

D-cyclohexane + Fe 3 

O 4 

/ MA (ϕ m ~ 1%) 


I(q), cm -1 

10 

1 

0.1 

fit by model “non-interacting 

polydisperse core-shell particles” 

F 2 N 

F 2 M 

Guinier 

approximations 

R g 

=8.0 nm 

R g 

=5.3 nm 

0.1 1 

q, nm -1 

I(q), cm -1 

10 

1 

0.1 

0.01 

1E-3 

R g 

=3.7 nm 

0.1 1 

q, nm -1 

R g 

=4 nm 

In contrast to correlation in particle locations, correlation in orientation of particle 

magnetic moments because of dipole-dipole interaction keeps its influence at small 

particle concentrations 

F 2 N 

F 2 M


Surface Effects. Spin canting 

• Labaye, Y., Crisan, O., Berger, L., et al., J. Appl. Phys. 91 (2002) 8715-8717


Surface Effects. Spin canting 

In a particle of radius 4 nm, about 50% of atoms lie on the surface. 

The concept of a well-defined super-moment breaks down. 

In real particles, the surface region thickness is very sensitive to particle shape 

distortion, surface roughness, surface impurities, defects (like vacancies), 

compositional inhomogeneity, surface chemical bonds with environment, etc. 

• Krycka, K.L., Booth, R.A., Hogg, C.R., et al., PRL 104 (2010) 207203

SANS on non-magnetized MFs: 

direct modeling 

D-solvents 

H-solvents 

I ( q) 

≈ (4 / 3) 

D n 

( R) 

2 

2 2 

3 

3 

2 

I( 

q) 

= (4 / 3) π n∫[( 

ρ − ρ ) R Φ( 

qR) 

+ ( ρ − ρ )( R + h) 

Φ( 

q( 

R + h))] 

D ( R dR + 

0 

∫ 

1 

2 2 2 

6 2 

+ (4 / 3) π nρ 

( R −δ 

) Φ ( q( 

R −δ 

)) D ( R dR 

∫ 

2 

2 6 2 

π n( 

ρ0 

− ρs ) R Φ ( qR) 

Dn 

( R) 

dR 

1 ⎛ ln 

exp 

⎜ − 

RS 2π 

⎝ 

= 

2 

2 

( R 

2S 

R ⎞ 

0) 

⎟ 

⎠ 

m n 

) 

100 

10 

1 

s n 

) 

Magnetite (ϕ m ~ 2 %) / OA / benzene 

H-benzene H-бензол 

D-benzene D-бензол 


Fitting parameters: 

• SLDs of the components (up to 4), 

• parameters of size distribution function (up to 4) 

• thicknesses of surfactant and non-magnetic 

layers (2) 

• concentration, background (2) 

Contrast variation is independent method 

for check-up multiparameter fits, especially 

for aggregated systems! 

I(q), см -1 

1 

0.1 

0.01 

1E-3 

D N 

(R) 

0.04 

0.02 

SANS МУРН 

0.00 

0 2 4 6 8 10 

• Avdeev, M.V., Balasoiu, M., Torok, Gy., et al., J. Magn. Magn. Mater. 252 (2002) 86-88 

• Aksenov, V.L., Avdeev, M.V., Balasoiu, M., et al., Appl. Phys. A 74 (2002) S943-S944 

R, нм 

TEM ПЭМ 

0.1 1 

q, нм -1

Summary 

• Small-angle neutron scattering is a powerful noninvasive structural 

technique, which gives information about particle sizes and shapes and 

interaction between them. 

• Basic functions approach for SANS experiments on a system of 

polydisperse particles has been developed; it shows specific differences 

comparing with the classical case of monodisperse particles; despite 

more complicated equations in general, still several expressions 

suggest transparent interpretation. 


• Compared to the monodisperse case, the contrast dependence of 

scattering integral parameters for the polydisperse system contains a 

number of additional parameters comprising information about 

polydispersity function of the system; their analysis increases the 

reliability of experimental results. 

• Magnetic scattering effect can be treated in terms of the basic functions 

approach, which provides an additional way to separate information 

about features of nuclear and magnetic structures of particles.

Acknowledgements 


M.V. Avdeev, 

V.L. Aksenov 

L.A. Bulavin 

V.I .Petrenko 

L. Vekas 

D. Bica [1952-2008] 

O. Marinica 

V.M. Garamus 

R. Willumeit 

E. Dubois 

R. Perzynski 

FLNP JINR, Russia 

Physics Department KNU, Ukraine 

Laboratory of Magnetic Fluids CFATR, Romania 

HZG, Germany 

UPMC, France 

Co-operation programs FLNP – KNU 

JINR – Hungarian Academy of Sciences, 

JINR – Romania 

RFBR-Helmholtz Germany, Program “Joint Research Groups”

Thank you for attention!

Small-Angle Neutron Scattering - Kfki

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