Small-Angle Neutron Scattering - Kfki
Small-Angle Neutron Scattering - Kfki
Small-Angle Neutron Scattering - Kfki
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<strong>Small</strong>-<strong>Angle</strong> <strong>Neutron</strong> <strong>Scattering</strong><br />
Mitglied in der Helmholtz-Gemeinschaft<br />
1<br />
Jülich Centre for <strong>Neutron</strong> Science<br />
Forschungszentrum Jülich GmbH<br />
2<br />
Frank Laboratory of <strong>Neutron</strong> Physics<br />
Joint Institute for Nuclear Research<br />
29. May 2013 | Artem Feoktystov 1 , Mikhail Avdeev 2
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Plan<br />
1. <strong>Small</strong>-angle neutron scattering<br />
2. Contrast variation. Basic functions<br />
3. Theory<br />
4. Core-shell structure<br />
5. Magnetic fluids<br />
6. SANS of polarized neutrons<br />
7. Spin canting<br />
8. Direct modeling<br />
9. Conclusions
<strong>Small</strong>-angle neutron scattering (SANS)<br />
Non-polarized neutrons<br />
= 0<br />
k<br />
qsinφ<br />
qcosφ<br />
k r k r<br />
0<br />
θ<br />
q r<br />
r<br />
q = k r<br />
−k r<br />
0<br />
4π<br />
θ<br />
q=<br />
sin<br />
λ 2<br />
θ <<br />
10<br />
o<br />
k 0<br />
For diluted samples (ϕ m < 3 vol. %)<br />
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H r )<br />
Differential<br />
cross section<br />
dσ<br />
d<br />
Ω Ω<br />
=<br />
dσ<br />
(q<br />
dΩ<br />
dσ<br />
( q)<br />
dΩ<br />
2<br />
≈ FN ( q)<br />
+<br />
Nuclear formfactor<br />
2<br />
3<br />
F<br />
2<br />
M<br />
( q)<br />
Magnetic formfactor
SAS: obtained information<br />
<strong>Scattering</strong> length density (SLD)<br />
profile<br />
Particle structure<br />
(form-factor F(q))<br />
I(q)~F 2 (q)S(q)<br />
<strong>Scattering</strong><br />
curve<br />
q<br />
Radial distribution function (RDF)<br />
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Particle interaction<br />
(structure-factor S(q))
<strong>Small</strong>-<strong>Angle</strong> <strong>Neutron</strong> <strong>Scattering</strong> (SANS):<br />
modern research objects<br />
Biological macromolecular complexes<br />
Self-organization in solutions of<br />
surfactants and lipids, complex liquids<br />
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Liquid dispersions of organic and<br />
magnetic materials<br />
Polymers<br />
Tendency – study of complex mixed systems!<br />
Features of neutron scattering:<br />
•wide abilities of isotopic hydrogendeuterium<br />
substitution<br />
•magnetic scattering<br />
•high penetration depth (absence of<br />
special sample preparation)
<strong>Neutron</strong> and X-ray scattering lengths<br />
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Contrast variation: basic idea<br />
“Core-Shell” particles<br />
in “Solvent”<br />
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Matching of scattering<br />
from “Shell”<br />
Matching of scattering<br />
from “Core”<br />
In SANS the realization is made by substitution of H with D in solvent!
Monodisperse non-magnetic systems:<br />
basic functions approach in contrast variation<br />
Contrast<br />
∆ρ<br />
=<br />
ρ −<br />
ρ s<br />
H.B. Stuhrmann (1975)<br />
average scattering length<br />
density of the particle<br />
scattering length<br />
density of the solvent<br />
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shape scattering<br />
scattering from density fluctuation<br />
r r<br />
ρ ( ) = ρ(<br />
) − ρ<br />
f
Example: core-shell particles<br />
ρ 1<br />
, R 1<br />
ρ<br />
I ( q)<br />
= n[(<br />
ρ ρs ρ ρ Φ qR<br />
2<br />
1<br />
− ) V1Φ(<br />
qR1<br />
) − (<br />
1<br />
−<br />
0)<br />
V0<br />
(<br />
0)]<br />
Φ( x)<br />
= 3(sin x − x cos x) / x<br />
n is the particle number density<br />
3<br />
V<br />
i<br />
=<br />
3<br />
( 4 / 3) πRi<br />
ρ 0<br />
, R 0<br />
ρ s<br />
ρ =<br />
V<br />
V<br />
V<br />
( ρ<br />
0 0<br />
ρ0<br />
+ 1−<br />
)<br />
1<br />
V1<br />
1<br />
∆ρ<br />
=<br />
ρ −<br />
ρ s<br />
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I( q) = nV ( ρ − ρ ) [ Φ( qR ) − Φ ( qR )] +<br />
2 2 2<br />
0 1 0 1 0<br />
+ 2 nVV ( ρ − ρ ) Φ( qR )[ Φ( qR ) − Φ( qR )]( ∆ ρ)<br />
+<br />
1 0 1 0 1 1 0<br />
+ nV Φ ( qR )( ∆ρ)<br />
2 2 2<br />
1 1<br />
Is<br />
Ics<br />
( q)<br />
( q)<br />
I ( ) c<br />
q
Core-shell particles. Basic functions<br />
R 0 = 4.5 nm<br />
R 1 = 6.0 nm<br />
ρ 0 = 7.0×10 10 cm -2<br />
ρ 1 = 0 cm -2<br />
ρ s1 = 5.4×10 10 cm -2<br />
ρ s2 = 3.0×10 10 cm -2<br />
ρ s3 = 1.0×10 10 cm -2<br />
5<br />
I 1<br />
(q)<br />
100<br />
10<br />
I c<br />
(q)<br />
4<br />
1<br />
I 3<br />
(q)<br />
0.1<br />
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I(q), arb. units<br />
3<br />
2<br />
1<br />
0<br />
I 2<br />
(q)<br />
0.01<br />
1E-3<br />
0.01 0.1 1 10<br />
q, nm -1 1E-3 0.01 0.1<br />
0<br />
I<br />
I s<br />
(q)<br />
cs<br />
(q)<br />
0.01 0.1 1<br />
q, nm -1
Guinier approximation<br />
Guinier approximation, 1939<br />
ellipsoid<br />
sphere<br />
cylinder<br />
I( q ~ 0) = I( 0)<br />
e −<br />
q<br />
< 1<br />
R<br />
g<br />
2 2<br />
q R g<br />
3<br />
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R g<br />
( )<br />
2<br />
I( 0) = nV ρ − ρs<br />
2<br />
– scattering intensity in zero angle<br />
– radius of gyration of scattering density distribution in particle
Monodisperse systems: Guinier regime<br />
1 2 2<br />
I( q ~ 0) = I( 0)exp( − q R g<br />
)<br />
Forward scattering<br />
intensity<br />
q<br />
< 1<br />
3 Rg<br />
Radius of gyration<br />
2 2<br />
I( 0 ) = nV ( ∆ρ)<br />
R = R + α ∆ρ − β ( ∆ρ)<br />
c<br />
2 2 2<br />
g c<br />
I(0)<br />
R 2 g<br />
radius of gyration<br />
of the shape<br />
2<br />
R c<br />
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0<br />
match point<br />
∆ρ<br />
0<br />
(∆ρ) −1
Radius of gyration:<br />
monodisperse systems<br />
R = R + α ∆ρ − β ( ∆ρ)<br />
2 2 2<br />
g c<br />
α = V<br />
−1<br />
c<br />
∫<br />
V c<br />
r<br />
ρ ( r )<br />
f<br />
r 2<br />
r<br />
d<br />
β =<br />
∫ ∫<br />
−2<br />
Vc<br />
f<br />
r1<br />
)<br />
f<br />
( r2<br />
)( r1<br />
r2<br />
) dr1<br />
dr2<br />
V V<br />
c<br />
c<br />
ρ<br />
( ρ<br />
β = 0<br />
β > 0<br />
α > 0<br />
α < 0<br />
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“light”<br />
component<br />
“heavy”<br />
component
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Example: ribosome<br />
70S ribosome<br />
1 2<br />
50S<br />
subunit<br />
30S<br />
subunit<br />
Proteins + RNA
Contrast variation on ribosome<br />
30S subunit<br />
is deuterated<br />
both 50S and 30S<br />
subunits are<br />
deuterated<br />
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50S ribosome<br />
H.Stuhrmann, et al., PNAS 73 (1976) 2379<br />
70S ribosome<br />
M.Koch, et al., Biophys. Struct. Mechan.<br />
4 (1978) 251
Systems with high polydispersity<br />
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• Avdeev, M.V ©<br />
What information one can obtain in general case from the SANS<br />
contrast variation?
Polydisperse non-magnetic systems<br />
I(<br />
q)<br />
= ∑ ni{<br />
I<br />
i<br />
i<br />
s<br />
( q)<br />
i<br />
+ ∆ ρI<br />
( q)<br />
+<br />
i<br />
cs<br />
( ∆<br />
i<br />
ρ)<br />
2<br />
I<br />
i<br />
c<br />
( q)}<br />
Size polydispersity<br />
Structural polydispersity<br />
ρ 1<br />
ρ 2<br />
ρ 1<br />
ρ 2<br />
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ρ 3<br />
ρ s<br />
ρ 3<br />
ρ s
Polydisperse magnetic systems<br />
I(<br />
q)<br />
i<br />
i<br />
2 i<br />
= ∑ ni { I<br />
s<br />
( q)<br />
+ ∆iρIcs<br />
( q)<br />
+ ( ∆iρ)<br />
Ic<br />
( q)}<br />
+<br />
i<br />
I<br />
m<br />
( q)<br />
Size polydispersity<br />
Structural polydispersity<br />
ρ 1<br />
ρ 2<br />
ρ 1<br />
ρ 2<br />
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ρ 3<br />
ρ s<br />
ρ 3<br />
ρ s
Polydisperse systems. General case<br />
I(q) = + + <br />
is averaging over the particle polydispersity function.<br />
The idea of the following transformations is to introduce the effective mean<br />
scattering length density, ρ e , independent of the averaging, so that the<br />
intensity equation takes the classical form but with the modified contrast.<br />
∆ ~ ρ =<br />
ρ e<br />
− ρ s<br />
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∆ ρ ρ − ρ + ρ − ρ = ρ − ρ + ∆<br />
~ ρ<br />
I(<br />
q)<br />
=<br />
=<br />
e e s<br />
e<br />
~<br />
I<br />
s<br />
( q)<br />
+ ∆<br />
~~<br />
ρI<br />
cs<br />
( q)<br />
+<br />
( ∆<br />
~ ρ)<br />
2<br />
~<br />
I<br />
c<br />
( q)
Modified basic functions<br />
I<br />
~ ( q)<br />
2<br />
s<br />
s<br />
e cs<br />
e<br />
~<br />
I<br />
~<br />
I<br />
cs<br />
c<br />
( q)<br />
( q)<br />
=< I ( q)<br />
> + < ( ρ − ρ ) I ( q)<br />
> + < ( ρ − ρ ) I ( q)<br />
=< I ( q)<br />
> + 2 < ( ρ − ρ ) I ( q)<br />
cs<br />
=< I ( q)<br />
c<br />
><br />
e<br />
c<br />
><br />
c<br />
><br />
Choice of<br />
ρe<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
ρ<br />
e<br />
=<<br />
∂I<br />
(0)<br />
∂ρ<br />
s<br />
= 0<br />
effective match point<br />
2<br />
ρI<br />
c<br />
( 0) > / < I<br />
c<br />
(0) >=< ρVc<br />
> / < Vc<br />
2<br />
>
Modified shape basic function<br />
ρ s<br />
From three measurements with different :<br />
~<br />
I ( q)<br />
c<br />
= −{(<br />
ρs2 − ρs3)<br />
I1(<br />
q)<br />
+ ( ρs3<br />
− ρs<br />
1)<br />
I2(<br />
q)<br />
+ ( ρs<br />
1<br />
− ρs2)<br />
I3(<br />
q)}/<br />
/( ρ<br />
s1 − ρ<br />
s2<br />
)( ρ<br />
s2<br />
− ρ<br />
s3)(<br />
ρ<br />
s3<br />
− ρ<br />
s1)<br />
ρ e<br />
Knowledge of exact -value is not necessary<br />
to obtain shape scattering function!<br />
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~<br />
I (0) = n <<br />
~ 2 2 2<br />
R =< V R ><br />
c<br />
2<br />
c<br />
V c<br />
c<br />
c<br />
/<br />
><br />
< V<br />
2<br />
c<br />
>
Guinier invariants<br />
I (0) = n∆ % ρ < V > +<br />
2 2<br />
c<br />
2 2<br />
( ρ ρe )<br />
c m<br />
(0)<br />
+ n < − V > + I<br />
~<br />
R<br />
2<br />
g<br />
=<br />
2 2<br />
< Vc<br />
Rc<br />
><br />
(<br />
2<br />
< V ><br />
c<br />
+<br />
A<br />
~ −<br />
∆ρ<br />
B<br />
( ∆<br />
~ ρ)<br />
2<br />
) /(1 +<br />
D<br />
( ∆<br />
~ ρ)<br />
2<br />
)<br />
I(0)<br />
classical<br />
monodisperse systems<br />
(a)<br />
~ 2<br />
R g<br />
< V<br />
< V<br />
R<br />
2 2<br />
c c<br />
2<br />
c<br />
><br />
><br />
Mitglied in der Helmholtz-Gemeinschaft<br />
0<br />
effective<br />
match point<br />
∆ρ ~<br />
In comparison with monodisperse case there are possibilities<br />
for analyzing Guinier region around effective match point!<br />
−<br />
B /<br />
D<br />
0<br />
( ∆ρ~<br />
)<br />
−1
Polydisperse systems<br />
Intensity at zero angle<br />
~<br />
I<br />
~<br />
I<br />
~<br />
s<br />
I cs<br />
(0)<br />
(0)<br />
(0)<br />
=<br />
=<br />
=<br />
n<br />
n < ( ρ − ρ )<br />
0<br />
<<br />
2<br />
c<br />
V c<br />
><br />
e<br />
2<br />
V<br />
2<br />
c<br />
><br />
I(0)<br />
classical<br />
monodisperse systems<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
I (0)<br />
=<br />
+<br />
2<br />
n∆ρ~<br />
< V<br />
2<br />
c<br />
n < ( ρ − ρ )<br />
e<br />
> +<br />
2<br />
V<br />
2<br />
c<br />
><br />
Full matching is not possible!<br />
0<br />
effective<br />
match point<br />
∆ρ ~
Polydisperse systems<br />
Radius of gyration<br />
~<br />
R<br />
2<br />
g<br />
=<br />
2 2<br />
< Vc<br />
Rc<br />
><br />
(<br />
2<br />
< V ><br />
c<br />
+<br />
A<br />
~ −<br />
∆ρ<br />
B<br />
( ∆<br />
~ ρ)<br />
2<br />
) /(1 +<br />
D<br />
( ∆<br />
~ ρ)<br />
2<br />
)<br />
classical<br />
monodisperse systems<br />
A<br />
= 1 2<br />
c<br />
< V<br />
2<br />
2<br />
( < αV<br />
> + 2 < ( − ) > )(a)<br />
2 c<br />
ρ ρ<br />
e<br />
Vc<br />
R<br />
><br />
c<br />
1<br />
B =<br />
< V<br />
2<br />
c<br />
( < βV<br />
><br />
2<br />
c<br />
2<br />
− < ( ρ − ρ ) V<br />
> − < ( ρ − ρ ) αV<br />
e<br />
2<br />
c<br />
R<br />
2<br />
c<br />
e<br />
> )<br />
2<br />
c<br />
><br />
~ 2<br />
R g<br />
< V<br />
< V<br />
R<br />
2 2<br />
c c<br />
2<br />
c<br />
><br />
><br />
Mitglied in der Helmholtz-Gemeinschaft<br />
D<br />
< ( ρ − ρ )<br />
e<br />
=<br />
2<br />
< Vc<br />
2<br />
><br />
V<br />
2<br />
c<br />
><br />
In comparison with monodisperse case there are possibilities<br />
for analyzing Guinier region around effective match point!<br />
−<br />
B /<br />
D<br />
0<br />
( ∆ρ~<br />
)<br />
−1
Case of two kinds of particles<br />
ρ ,ε ,ε<br />
1 1<br />
2 2<br />
ε + ε<br />
1 2<br />
=<br />
1<br />
2<br />
2 2<br />
ρ = ( ε ρ V + ε ρ V ) /( ε V + ε V<br />
e<br />
2<br />
1 1 c1<br />
2 2 c2<br />
1 c1<br />
2 c2<br />
)<br />
I(0)<br />
=<br />
ρ ~ 2 2 2<br />
2 2 2 2 2<br />
n∆ρ ( ε V + ε V ) + n(<br />
ρ − ρ ) ε ε V V /( ε V + ε V )<br />
1<br />
c1<br />
2<br />
c2<br />
1<br />
2<br />
1<br />
2<br />
c1<br />
c2<br />
1<br />
c1<br />
2<br />
c2<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
Residual intensity at zero angle is determined by the difference in SLDs.
Structural polydispersity<br />
V c<br />
= const<br />
=< ρ ><br />
I<br />
ρ e<br />
2 2<br />
2 2 ~ 2 2 2 2<br />
( 0) = n∆<br />
~ ρ Vc<br />
+ n < ( ρ − ρe<br />
) > Vc<br />
= n∆ρ<br />
Vc<br />
+ nσ<br />
ρVc<br />
~ 2 2 A B<br />
Rg = ( Rc<br />
+ ~ − ~ ) /(1 +<br />
2<br />
∆ρ<br />
( ∆ρ)<br />
D<br />
( ∆<br />
~ ρ)<br />
2<br />
)<br />
A =< α ><br />
B<br />
=<<br />
β > − <<br />
2 2<br />
2 2<br />
( ρ − ρe<br />
) α > − < ( ρ − ρe<br />
) > Rc<br />
=< β > − < ( ρ − ρe<br />
) α > −σ<br />
ρ<br />
Rc<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
D =< ρ ρe<br />
>= σ<br />
2 2<br />
( − )<br />
ρ
Example: ferritin<br />
protein shell, R 1 = 6 nm,<br />
ρ 1 = 1.9-2.95 × 10 10 cm -2<br />
ρ s<br />
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ferrite core, R 0 = 4 nm<br />
SANS curves from<br />
iron unsaturated ferritin<br />
in various H 2 O/D 2 O mixtures<br />
H.Stuhrmann, E.Duee, JAC 8 (1975) 538
Mitglied in der Helmholtz-Gemeinschaft<br />
Density of the core is distributed with<br />
σ ρ = 0.35 × 10 10 cm −2 = −1.14 × 10 −3
For particle of two components with common center of mass:<br />
2<br />
2 ( ρ0<br />
− ρs<br />
)( V0<br />
/ V1<br />
) R0<br />
+ ( ρ1<br />
− ρs<br />
)(1 − ( V0<br />
/ V1<br />
)) R<br />
Rg<br />
=<br />
∆ρ<br />
2<br />
1<br />
I.N.Serdyuk, B.A.Fedorov,<br />
J. Polym. Sci. 11 (1973) 645<br />
2 2<br />
2 2 ( V0<br />
/ V1<br />
)(1 − ( V0<br />
/ V1<br />
))( ρ1<br />
− ρ0)(<br />
R1<br />
− R0<br />
)<br />
Rg<br />
= Rc<br />
+<br />
∆ρ<br />
=<br />
R<br />
2<br />
c<br />
+<br />
α<br />
∆ρ<br />
For iron unsaturated ferritin:<br />
~<br />
R<br />
= ( R<br />
< α ><br />
∆<br />
~ ρ<br />
B<br />
( ∆<br />
~ ρ)<br />
) /(1<br />
2 2<br />
g c<br />
+ − +<br />
2<br />
D<br />
( ∆<br />
~ ρ)<br />
2<br />
)<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
2<br />
< α<br />
>=<br />
( V0 / V1<br />
)(1 − ( V0<br />
/ V1<br />
))( ρ1− < ρ0<br />
> )( R1<br />
− R<br />
< ρ<br />
>=<br />
e<br />
( V<br />
ρ<br />
0<br />
/ V1<br />
) < ρ0<br />
> + (1 − ( V0<br />
/ V1<br />
))<br />
Analysis of B, D<br />
1<br />
2<br />
0<br />
)<br />
< ρ<br />
< ρ<br />
10 -2<br />
0 >=<br />
5 .0×<br />
10 cm<br />
10 -2<br />
0 >=<br />
4 .9×<br />
10 cm<br />
additional data for σ ρ
Magnetic neutron scattering<br />
Cross-section of the magnetic moment<br />
of one atom in domain:<br />
dσ<br />
dΩ<br />
2 2<br />
( ) = m<br />
vm<br />
p<br />
p - magnetic scattering length<br />
vm<br />
= γ −η ( ηγ )<br />
- magnetic interaction vector<br />
Nuclear + magnetic scattering:<br />
dσ<br />
=<br />
dΩ<br />
b<br />
2<br />
+<br />
2 2<br />
2bp(<br />
vm t ) + vm<br />
p<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
unit vector along the<br />
magnetic moment<br />
direction<br />
ν m<br />
=sinα, α - angle between<br />
unit vector along<br />
the scattering vector<br />
γ<br />
and<br />
q<br />
neutron spin<br />
non-polarized beam:<br />
dσ<br />
=<br />
dΩ<br />
b<br />
disoriented domains:<br />
dσ<br />
dΩ<br />
2<br />
+ v<br />
2<br />
m<br />
p<br />
b<br />
2 v<br />
2 p<br />
2 b<br />
2<br />
= + <<br />
m<br />
><br />
Ω<br />
= +<br />
2<br />
2 p<br />
3<br />
2
Effect of magnetic scattering<br />
Monodisperse case<br />
∆<br />
~ ρ = ∆ρ<br />
~<br />
Ic ( q)<br />
= Ic(<br />
q)<br />
~<br />
I<br />
s<br />
( q)<br />
= Is<br />
( q)<br />
+ I<br />
~<br />
I ( q)<br />
I ( q)<br />
cs<br />
=<br />
cs<br />
m<br />
( q)<br />
I<br />
I<br />
=<br />
2<br />
3<br />
ρ<br />
2 2<br />
m<br />
( 0) nV m m<br />
m<br />
( q<br />
~ 0) =<br />
I<br />
m<br />
(0)(1 −<br />
R<br />
2 q 2<br />
m<br />
/ 3)<br />
~<br />
I<br />
(0)<br />
=<br />
2<br />
3<br />
2<br />
s<br />
nV m<br />
ρ<br />
2<br />
I<br />
s<br />
( q ~ 0) = I<br />
m<br />
(0) −{<br />
−βnVc<br />
+<br />
2<br />
m<br />
~ 2<br />
I (0) R } q<br />
2<br />
m m<br />
/ 3<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
~<br />
R<br />
2<br />
g<br />
=<br />
( R<br />
2<br />
c<br />
+<br />
α<br />
−<br />
∆ρ<br />
I<br />
∆ρ +<br />
2<br />
3<br />
ρ<br />
2 2<br />
2 2<br />
( 0) = n Vc<br />
n<br />
mVm<br />
2<br />
β − (2 / 3) ρ<br />
m<br />
( V<br />
( ∆ρ)<br />
2<br />
m<br />
2<br />
/ V<br />
2<br />
c<br />
) R<br />
2<br />
m<br />
) /(1 +<br />
2<br />
(2 / 3) ρ<br />
m<br />
( V<br />
( ∆ρ)<br />
2<br />
m<br />
2<br />
/ V<br />
2<br />
c<br />
)<br />
)
Effect of magnetic scattering<br />
Monodisperse case.<br />
Radius of gyration: homogeneous particles<br />
α = 0 β = 0<br />
R 2 g<br />
2 2 2 2<br />
~ 2 2 (2 / 3) ρm(<br />
Vm<br />
/ Vc<br />
) Rm<br />
R<br />
g<br />
= ( Rc<br />
+<br />
) /(1 +<br />
2<br />
( ∆ρ)<br />
a<br />
2<br />
(2 / 3) ρm(<br />
V<br />
( ∆ρ)<br />
2<br />
m<br />
2<br />
/ V<br />
2<br />
c<br />
)<br />
)<br />
The view depends on which radius is larger!<br />
R 2 m<br />
R m<br />
> R c<br />
If for spherical particles<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
R 2 c<br />
R 2 m<br />
0<br />
R m<br />
< R c<br />
(∆ρ) -1<br />
R<br />
m<br />
= R c<br />
~ 2 2<br />
g<br />
R c<br />
R =<br />
No magnetic scattering effect!
Effect of magnetic scattering<br />
Polydisperse case<br />
Size-polydisperse non-homogeneous particles<br />
2 2<br />
2 2<br />
2<br />
I ( 0) = n∆ρ~<br />
< Vc<br />
> + n < ( ρ − ρe)<br />
Vc<br />
> + (2 / 3) nρm<br />
< Vm<br />
2<br />
><br />
~<br />
R<br />
2<br />
g<br />
=<br />
2 2<br />
< Vc<br />
Rc<br />
><br />
(<br />
2<br />
< V ><br />
c<br />
+<br />
A<br />
~ −<br />
∆ρ<br />
B<br />
( ∆<br />
~ ρ)<br />
2<br />
) /(1 +<br />
D<br />
( ∆<br />
~ ρ)<br />
2<br />
)<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
B<br />
1 2<br />
2<br />
A = ( < αV<br />
> + 2 < ( − )<br />
2 > )<br />
2 c<br />
ρ ρ<br />
e<br />
Vc<br />
R<br />
< V ><br />
c<br />
c<br />
= 1 2<br />
2<br />
2 2 2<br />
2 2<br />
( < βV<br />
> − < ( − ) > − < ( − ) > −(2 / 3) <<br />
2 ><br />
2 c<br />
ρ ρe<br />
αVc<br />
ρ ρe<br />
Vc<br />
Rc<br />
ρm<br />
Vm<br />
R<br />
< V ><br />
m<br />
c<br />
2 2<br />
2 2<br />
< ( ρ − ρe)<br />
Vc<br />
> (2 / 3) ρm<br />
< Vm<br />
><br />
D = +<br />
2<br />
2<br />
< Vc<br />
> < Vc<br />
><br />
)
Basic functions approach<br />
ρ =<br />
const<br />
bi<br />
ρ = ∑V<br />
i<br />
ρ ≠<br />
const<br />
ρ<br />
e<br />
2<br />
< ρV<br />
><br />
= <<br />
2<br />
V ><br />
Contrast<br />
Modified contrast<br />
ρs<br />
∆ ρ = ρ − ρ s<br />
ρs<br />
∆ % ρ = ρ − ρ<br />
e<br />
s<br />
I( q) = I ( q) + ∆ ρI ( q) + ( ∆ρ) 2 I ( q)<br />
I( q) = I % ( q) + ∆ % ρI % ( q) + ( ∆ % ρ) 2 I % ( q)<br />
s cs c<br />
s cs c<br />
∆ ρ = ∞ ∆ % ρ = ∞<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
I ( ) c<br />
q<br />
• Stuhrmann, H.B.(1982). <strong>Small</strong>-<strong>Angle</strong> X-ray <strong>Scattering</strong>, edited by O. Glatter & O. Kratky, pp.197-213, London: Academic Press<br />
• Avdeev, M.V. J. Appl. Cryst. 40 (2007) 56-70<br />
I%<br />
( q) =< I ( q)<br />
><br />
c<br />
c
Integral scattering parameters<br />
2 2<br />
2 2 2 2<br />
I( 0 ) = nV ( ∆ρ)<br />
I( 0) = n < V > ( ∆ % ρ) + n < ( ρ − ρ ) V > +<br />
c<br />
<strong>Scattering</strong> intensity in zero angle:<br />
c e c<br />
2 2<br />
+ ( 2 3)<br />
nρ<br />
< V ><br />
m<br />
m<br />
I(0)<br />
I(0)<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
0 ∆ρ<br />
0 ∆ρ
Integral scattering parameters<br />
R<br />
α<br />
= R + −<br />
∆ρ<br />
2 2<br />
g c<br />
Radius of gyration:<br />
2 2<br />
β<br />
2<br />
⎛ < Vc<br />
Rc<br />
> A B ⎞ ⎛ D ⎞<br />
R<br />
2 g<br />
= ⎜ + − ⎟ +<br />
( ∆ρ)<br />
Vc<br />
ρ ( ρ) ⎜1<br />
2 2 2<br />
< > ∆ ∆ ( ∆ρ)<br />
⎟<br />
⎝<br />
% % ⎠ ⎝ % ⎠<br />
α0<br />
R 2 g<br />
R 2 g A
Magnetic fluids (ferrofluids)<br />
Mitglied in der Helmholtz-Gemeinschaft
Magnetic fluid: biomedical applications<br />
The main direction is cancer treatment!<br />
Drug delivery<br />
Therapy<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
Diagnostics
Magnetic drug concentrator:<br />
idea (SIEMENS)<br />
Mitglied in der Helmholtz-Gemeinschaft
Magnetic drug concentrator:<br />
prototype (SIEMENS)<br />
Mitglied in der Helmholtz-Gemeinschaft
Magnetic fluids (ferrofluids):<br />
specific properties for biomedicine<br />
Magnetic nanoparticles, size d = 2-30 nm<br />
(one-domain magnetic state →<br />
magnetic moment of single particle ~10 3 -10 5 µ B<br />
→ superparamagnetic behavior)<br />
Liquid carrier<br />
Surfactant shell,<br />
thickness d = 1-2 nm<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
R.E. Rosensweig, 1966<br />
<strong>Small</strong> size – large specific surface,<br />
natural removal from organisms<br />
Combination of liquidity<br />
and magnetism are widely<br />
used in technical applications<br />
Magnetism – control by external<br />
magnetic field
Classical stabilization procedure for<br />
nonpolar organic magnetic fluids<br />
Co-precipitation<br />
aqueous solutions Fe 2+ , Fe 3+ with NH 4 OH (T = 80 o C)<br />
Oleic Acid (OA)<br />
Oleic Acid (OA)<br />
Coating<br />
chemisorption of oleic acid (T = 80 o C)<br />
Nanomagnetite<br />
Washing<br />
water<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
Flocculation<br />
acetone<br />
Initial dispersing<br />
light hydrocarbon<br />
Repeating flocculation and re-dispersing<br />
surfactant removal and particle concentrating<br />
Organic liquid<br />
(hydrocarbon)<br />
Organic non-polar magnetic fluids are<br />
often the first step in synthesis of<br />
various biocompatible systems
Types of ferrofluid stabilization<br />
Single sterical<br />
stabilization<br />
Double sterical<br />
stabilization<br />
Ionic (electrostatic)<br />
stabilization<br />
H +<br />
H +<br />
H +<br />
H +<br />
H + H + H + H +<br />
H<br />
H + +<br />
H + H +<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
Surfactant chemisorption;<br />
no surfactant excess<br />
I layer – surfactant chemisorption;<br />
II layer – physical adsorption of<br />
surfactant in excess<br />
Adsorption of ions<br />
(H + , ОH − , citrate-ion)<br />
(Massart’s method)
Surfactants for stabilization<br />
of magnetic fluids based on organic liquids<br />
Unsaturated monocarboxylic acid<br />
Kinked<br />
double bond<br />
Oleic Acid (OA)<br />
СН 3 (CH 2 ) 8 =(СН 2 ) 8 СOОН or C 18 H 34 O 2<br />
Excellent stabilizer<br />
Saturated monocarboxylic acids<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
Stearic Acid (SA)<br />
СН 3 (CH 2 ) 16 СOОН or C 18 H 36 O 2<br />
Extremely bad stabilizer<br />
Palmitic Acid (PA)<br />
C 16 H 32 O 2<br />
Myristic Acid (MA)<br />
C 14 H 28 O 2<br />
Lauric Acid (LA)<br />
C 12 H 24 O 2<br />
Intermediate<br />
case
Nuclear and magnetic particle form-factors<br />
Nuclear scattering<br />
Magnetic scattering<br />
F<br />
2<br />
( q)<br />
= [<br />
R<br />
max<br />
∫<br />
0<br />
sin( qr)<br />
( ρ(<br />
r)<br />
− ρs<br />
)<br />
qr<br />
4πr<br />
2<br />
dr]<br />
2<br />
F<br />
2<br />
N<br />
( q)<br />
= [( ρ0<br />
- ρ1)<br />
V0Φ(<br />
qR0<br />
) + ( ρ1<br />
- ρs<br />
) V1Φ(<br />
qR1<br />
)]<br />
2<br />
F<br />
2<br />
M<br />
( q)<br />
= [ ρmVmΦ(<br />
qRm<br />
)]<br />
2<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
Model spherical “core-shell”<br />
Model “homogeneous sphere”<br />
Unmagnetized FF<br />
Φ( x)<br />
= 3(sin( x)<br />
− x cos( x)) / x<br />
I +<br />
2<br />
3<br />
2<br />
2<br />
( q)<br />
= nF N<br />
nFM<br />
3
Contrast variation on MF: principle stages<br />
Dissolution of initial concentrated system<br />
(ϕ m 1-10%) with mixtures of H- and D- carriers;<br />
measurements of scattering curves.<br />
Determination of<br />
effective match point<br />
from minimum of dependence I(0) ~ ρ s<br />
Minimization of<br />
χ<br />
2<br />
Experimental separation<br />
on basic functions<br />
1 [ I ( q) I ( q) I ( q) ( ) I ( q)]<br />
= ∑ % % % % %<br />
N − 3 ( )<br />
k<br />
2 2<br />
k<br />
−<br />
s<br />
− ∆ρk cs<br />
− ∆ρk c<br />
2<br />
σ<br />
k<br />
q<br />
I k<br />
(q), σ k<br />
(q) are experimental data for k-th contrast;<br />
N is number of contrasts.<br />
SLD map for MF components<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
Separation to modified basic functions<br />
Comparison of I s with scattering<br />
in effective match point<br />
(check-out separation)<br />
Analysis of I c .<br />
Comparison of I c (total shape)<br />
with scattering in H-carrier<br />
(only «sphere» of magnetic material)<br />
7.0×10 10 cm -2<br />
2.0×10 10 cm -2 D-solvent<br />
1.0×10 10 cm -2<br />
0.0×10 10 cm -2<br />
magnetic SLD<br />
Н-solvent<br />
magnetic<br />
materials<br />
surfactants
Contrast variation on FFs:<br />
separate nonhomogeneous particles<br />
Intensity, cm -1<br />
10<br />
1<br />
0.1<br />
0.01<br />
10<br />
Magnetite (ϕ m ~ 1 %) coated with oleic (OA) or myristic (MA) acids in benzene<br />
OA<br />
10% C 6<br />
D 6<br />
20% C 6<br />
D 6<br />
30% C 6<br />
D 6<br />
50% C 6<br />
D 6<br />
55% C 6<br />
D 6<br />
60% C 6<br />
D 6<br />
70% C 6<br />
D 6<br />
75% C 6<br />
D 6<br />
90% C 6<br />
D 6<br />
q, nm -1<br />
0.1 1<br />
100% C 6<br />
D 6<br />
100% C 6<br />
D 6<br />
90% C 6<br />
D 6<br />
75% C 6<br />
D 6<br />
60% C 6<br />
D 6<br />
I (0)<br />
2<br />
R g<br />
I(0) for OA, cm -1<br />
I(0) для ОК, см -1<br />
35<br />
30<br />
25<br />
20<br />
15<br />
10<br />
5<br />
0<br />
80<br />
70<br />
60<br />
МК<br />
MA<br />
34.2 %<br />
ОК<br />
OA<br />
63 %<br />
0<br />
0.0 0.2 0.4 0.6 0.8 1.0 1.2<br />
ОК<br />
OA<br />
МК<br />
MA<br />
[C 6<br />
D 6<br />
]/[C 6<br />
D 6<br />
+C 6<br />
H 6<br />
]<br />
10<br />
8<br />
6<br />
4<br />
2<br />
I(0) для for MA, МК, см cm -1 -1<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
Intensity, cm -1<br />
1<br />
0.1<br />
0.01<br />
MA<br />
0.1 1<br />
q, nm -1<br />
50% C 6<br />
D 6<br />
40% C 6<br />
D 6<br />
38% C 6<br />
D 6<br />
35% C 6<br />
D 6<br />
32% C 6<br />
D 6<br />
30% C 6<br />
D 6<br />
20% C 6<br />
D 6<br />
10% C 6<br />
D 6<br />
R<br />
2<br />
R 2 g ,<br />
g<br />
nm<br />
2<br />
,нм2<br />
50<br />
40<br />
30<br />
20<br />
10<br />
0<br />
-10<br />
-18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10<br />
(∆ρ) -1 −1 ×10 -10 ,,cm см 22<br />
/
Contrast variation on FFs:<br />
separate nonhomogeneous particles<br />
Separation on modified basic functions<br />
Comparison of I s (q) with the scattering<br />
at the effective match point<br />
1.1<br />
1.0<br />
0.9<br />
0.8<br />
0.7<br />
Ic<br />
1<br />
MA<br />
MA 35% C 6<br />
D 6<br />
OA<br />
OA 60% C 6<br />
D 6<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
Ics<br />
Is<br />
I s<br />
, cm -1<br />
0.1<br />
0.1<br />
0.0<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
-0.1<br />
0.1 1<br />
q, nm -1<br />
0.1 1<br />
q, nm -1
Contrast variation on FFs:<br />
separate non-homogeneous particles<br />
Magnetite (ϕ m ~ 1 %) coated with oleic (OA) or myristic (MA) acids in benzene<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
I I c<br />
x см c × 10 -20 , cm 3<br />
Analysis of I с (q) and comparison with<br />
scattering from magnetite component<br />
10<br />
1<br />
0.1<br />
0.01<br />
1E-3<br />
1E-4<br />
D N<br />
(R)<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
from Н-solvent<br />
MA<br />
МК<br />
OA<br />
ОК<br />
0.0<br />
0 2 4 6 8 10<br />
R, нм<br />
ОК<br />
МК<br />
from I c (q)<br />
q, нм nm -1 -1<br />
0.1 1<br />
OA<br />
MA<br />
I c (q)<br />
I(q)<br />
at 0% C 6 D 6<br />
Size and polydispersity decrease when substituting OA with MA!<br />
Thickness of effective surfactant shell (1.5 nm) does not change with substitution!<br />
• Feoktystov, A.V., Avdeev, M.V., Aksenov, V.L., Bulavin, L.A., et al., Surface Investigations 3 (2009) 1-4<br />
• 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<br />
Intensity, cm -1<br />
100<br />
10<br />
1<br />
0.1<br />
magnetite/lauric acid/water<br />
0% D 2<br />
O<br />
10% D 2<br />
O<br />
20% D 2<br />
O<br />
30% D 2<br />
O<br />
40% D 2<br />
O<br />
50% D 2<br />
O<br />
60% D 2<br />
O<br />
70% D 2<br />
O<br />
80% D 2<br />
O<br />
90% D 2<br />
O<br />
Intensity, cm -1<br />
10<br />
1<br />
0.1<br />
magnetite/myristic acid/water<br />
0% D 2<br />
O<br />
10% D 2<br />
O<br />
20% D 2<br />
O<br />
30% D 2<br />
O<br />
40% D 2<br />
O<br />
50% D 2<br />
O<br />
60% D 2<br />
O<br />
70% D 2<br />
O<br />
80% D 2<br />
O<br />
90% D 2<br />
O<br />
0.01<br />
0.1 1<br />
q, nm -1<br />
0.1 1<br />
q, nm -1<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
I( q ~ 0) = I( 0)<br />
e −<br />
2 2<br />
q R g<br />
3<br />
Due to a strong aggregation in both samples<br />
the Guinier region is inaccessible!<br />
• Feoktystov, A.V., Bulavin, L.A., Avdeev, M.V., et al., Ukr. J. Phys. 54 (2009) 266-273
Dependence of match point on q<br />
Match Point, D 2<br />
O vol. %<br />
60<br />
50<br />
40<br />
30<br />
20<br />
10<br />
LA<br />
MA<br />
0<br />
0.0 0.2 0.4 0.6 0.8 1.0<br />
q, nm -1<br />
Mitglied in der Helmholtz-Gemeinschaft<br />
In case of myristic acid stabilization homogeneous aggregates with the size<br />
bigger than 20 nm are present in the system.<br />
In magnetic fluid with lauric acid stabilization the aggregates in the system<br />
are inhomogeneous.
Structure parameters of aggregates<br />
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I c<br />
, x10 -20 cm 3<br />
10<br />
1<br />
0.1<br />
0.01<br />
1E-3<br />
0.1 1<br />
q, nm -1<br />
LA<br />
MA<br />
p(r)<br />
0.15<br />
0.10<br />
0.05<br />
Characteristic radii of aggregates:<br />
(19.5 ± 0.3) nm (LA+LA stabilization)<br />
(13.8 ± 0.1) nm (MA+MA stabilization)<br />
0.25 TEM of separate particles<br />
0.20<br />
Indirect Fourier Transform of I c<br />
(q)<br />
MA<br />
Indirect Fourier Transform of scattering<br />
curve of the sample on H 2<br />
O<br />
LA<br />
MA<br />
LA<br />
0.00<br />
0 10 20 30 40 50<br />
r, nm<br />
In case of lauric acid stabilization nanomagnetite is coated with<br />
a surfactant shell with thickness ~3.5 nm.<br />
In the sample with myristic acid stabilization the aggregates in the system<br />
consist of nanomagnetite with one incomplete layer of the acid.
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Aggregate structure of both ferrofluids<br />
magnetite/lauric acid/water<br />
magnetite/myristic acid/water
Contrast variation on FFs:<br />
separate homogeneous particles<br />
Biocompatible FFs with maghemite coated by citric ions (0.2 nm) in water.<br />
Coulomb interaction compensated (NaCl addition). Magnetic interaction compensated (ϕ m ~ 1%).<br />
I c (q)<br />
(negligible magnetic scattering)<br />
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I(q), cm -1<br />
100<br />
10<br />
1<br />
0.1<br />
0.01<br />
1E-3<br />
1E-4<br />
0 % D2O<br />
70 % D2O<br />
80 % D2O<br />
90 % D2O<br />
95 % D2O<br />
100 % D2O<br />
I s (q)<br />
(matched nuclear scattering)<br />
0.1 1<br />
q, nm -1<br />
R g<br />
2<br />
110<br />
100<br />
90<br />
80<br />
70<br />
60<br />
50<br />
characteristic nuclear radius<br />
R N ~ 10 nm<br />
0 2 4 6 8 10<br />
∆ρ~<br />
−1<br />
( ) ×<br />
Increase in apparent particle size<br />
because of residual Van der Waals interaction<br />
(non-uniform surface charge distribution)<br />
• Avdeev, M.V., Dubois, E., Mériguet, G., et al., J. Appl. Cryst. 42 (2009) 1009-1019<br />
characteristic magnetic radius<br />
R M ~ 7.2 nm<br />
10<br />
-10<br />
cm<br />
2
<strong>Small</strong>-angle neutron scattering<br />
Polarized neutrons – magnetic structure<br />
qsinφ<br />
H r ∞<br />
k<br />
qcosφ<br />
k r q r<br />
θ<br />
k r 0<br />
r<br />
q = k r<br />
−k r<br />
0<br />
4π<br />
θ<br />
q=<br />
sin2<br />
λ<br />
k 0<br />
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Incident beam of<br />
polarized neutrons with<br />
wavelength λ<br />
d<br />
dΩ<br />
−<br />
d<br />
dΩ<br />
Dissolved samples.<br />
Cross section for two states of neutron polarization<br />
+<br />
σ r 2<br />
2<br />
2<br />
( q)<br />
≈ FN<br />
( q)<br />
+ { FM<br />
( q)<br />
− 2FN<br />
( q)<br />
FM<br />
( q)}sin<br />
σ r 2<br />
2<br />
2<br />
( q)<br />
≈ FN ( q)<br />
+ { FM<br />
( q)<br />
+ 2FN<br />
( q)<br />
FM<br />
( q)}sin<br />
Nuclear formfactor<br />
Magnetic formfactor<br />
ϕ<br />
ϕ
<strong>Small</strong>-angle neutron scattering:<br />
polarized neutrons<br />
Magnetite (ϕ m ~ 1%) with OA and MA coating in D-cyclohexane<br />
I + ,I − , H=0 I + , H=H ∞ I − , H=H ∞<br />
OA<br />
2D pattern of neutron scattering for two states of<br />
neutron polarization: parallel (I − ) and antiparallel<br />
(I + ) to external magnetic field (2.5 Т) on<br />
samples:<br />
MA<br />
Detector size 55×55 cm 2<br />
Sample-detector distance 4.5 m<br />
<strong>Neutron</strong> wavelength 0.83 nm<br />
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H r ∞
Separation of nuclear and<br />
magnetic neutron scattering<br />
D-cyclohexane + Fe 3<br />
O 4<br />
/ OA (ϕ m ~ 1%)<br />
D-cyclohexane + Fe 3<br />
O 4<br />
/ MA (ϕ m ~ 1%)<br />
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I(q), cm -1<br />
10<br />
1<br />
0.1<br />
fit by model “non-interacting<br />
polydisperse core-shell particles”<br />
F 2 N<br />
F 2 M<br />
Guinier<br />
approximations<br />
R g<br />
=8.0 nm<br />
R g<br />
=5.3 nm<br />
0.1 1<br />
q, nm -1<br />
I(q), cm -1<br />
10<br />
1<br />
0.1<br />
0.01<br />
1E-3<br />
R g<br />
=3.7 nm<br />
0.1 1<br />
q, nm -1<br />
R g<br />
=4 nm<br />
In contrast to correlation in particle locations, correlation in orientation of particle<br />
magnetic moments because of dipole-dipole interaction keeps its influence at small<br />
particle concentrations<br />
F 2 N<br />
F 2 M
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Surface Effects. Spin canting<br />
• Labaye, Y., Crisan, O., Berger, L., et al., J. Appl. Phys. 91 (2002) 8715-8717
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Surface Effects. Spin canting<br />
In a particle of radius 4 nm, about 50% of atoms lie on the surface.<br />
The concept of a well-defined super-moment breaks down.<br />
In real particles, the surface region thickness is very sensitive to particle shape<br />
distortion, surface roughness, surface impurities, defects (like vacancies),<br />
compositional inhomogeneity, surface chemical bonds with environment, etc.<br />
• Krycka, K.L., Booth, R.A., Hogg, C.R., et al., PRL 104 (2010) 207203
SANS on non-magnetized MFs:<br />
direct modeling<br />
D-solvents<br />
H-solvents<br />
I ( q)<br />
≈ (4 / 3)<br />
D n<br />
( R)<br />
2<br />
2 2<br />
3<br />
3<br />
2<br />
I(<br />
q)<br />
= (4 / 3) π n∫[(<br />
ρ − ρ ) R Φ(<br />
qR)<br />
+ ( ρ − ρ )( R + h)<br />
Φ(<br />
q(<br />
R + h))]<br />
D ( R dR +<br />
0<br />
∫<br />
1<br />
2 2 2<br />
6 2<br />
+ (4 / 3) π nρ<br />
( R −δ<br />
) Φ ( q(<br />
R −δ<br />
)) D ( R dR<br />
∫<br />
2<br />
2 6 2<br />
π n(<br />
ρ0<br />
− ρs ) R Φ ( qR)<br />
Dn<br />
( R)<br />
dR<br />
1 ⎛ ln<br />
exp<br />
⎜ −<br />
RS 2π<br />
⎝<br />
=<br />
2<br />
2<br />
( R<br />
2S<br />
R ⎞<br />
0)<br />
⎟<br />
⎠<br />
m n<br />
)<br />
100<br />
10<br />
1<br />
s n<br />
)<br />
Magnetite (ϕ m ~ 2 %) / OA / benzene<br />
H-benzene H-бензол<br />
D-benzene D-бензол<br />
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Fitting parameters:<br />
• SLDs of the components (up to 4),<br />
• parameters of size distribution function (up to 4)<br />
• thicknesses of surfactant and non-magnetic<br />
layers (2)<br />
• concentration, background (2)<br />
Contrast variation is independent method<br />
for check-up multiparameter fits, especially<br />
for aggregated systems!<br />
I(q), см -1<br />
1<br />
0.1<br />
0.01<br />
1E-3<br />
D N<br />
(R)<br />
0.04<br />
0.02<br />
SANS МУРН<br />
0.00<br />
0 2 4 6 8 10<br />
• Avdeev, M.V., Balasoiu, M., Torok, Gy., et al., J. Magn. Magn. Mater. 252 (2002) 86-88<br />
• Aksenov, V.L., Avdeev, M.V., Balasoiu, M., et al., Appl. Phys. A 74 (2002) S943-S944<br />
R, нм<br />
TEM ПЭМ<br />
0.1 1<br />
q, нм -1
Summary<br />
• <strong>Small</strong>-angle neutron scattering is a powerful noninvasive structural<br />
technique, which gives information about particle sizes and shapes and<br />
interaction between them.<br />
• Basic functions approach for SANS experiments on a system of<br />
polydisperse particles has been developed; it shows specific differences<br />
comparing with the classical case of monodisperse particles; despite<br />
more complicated equations in general, still several expressions<br />
suggest transparent interpretation.<br />
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• Compared to the monodisperse case, the contrast dependence of<br />
scattering integral parameters for the polydisperse system contains a<br />
number of additional parameters comprising information about<br />
polydispersity function of the system; their analysis increases the<br />
reliability of experimental results.<br />
• Magnetic scattering effect can be treated in terms of the basic functions<br />
approach, which provides an additional way to separate information<br />
about features of nuclear and magnetic structures of particles.
Acknowledgements<br />
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M.V. Avdeev,<br />
V.L. Aksenov<br />
L.A. Bulavin<br />
V.I .Petrenko<br />
L. Vekas<br />
D. Bica [1952-2008]<br />
O. Marinica<br />
V.M. Garamus<br />
R. Willumeit<br />
E. Dubois<br />
R. Perzynski<br />
FLNP JINR, Russia<br />
Physics Department KNU, Ukraine<br />
Laboratory of Magnetic Fluids CFATR, Romania<br />
HZG, Germany<br />
UPMC, France<br />
Co-operation programs FLNP – KNU<br />
JINR – Hungarian Academy of Sciences,<br />
JINR – Romania<br />
RFBR-Helmholtz Germany, Program “Joint Research Groups”
Thank you for attention!<br />
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