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Steve Kline
NCNR Summer School
Neutron Small Angle Scattering andReflectometry from Submicron Structures
June 5 - 9 2000
Fundamentals ofSmall-Angle Neutron Scattering
• Generalization of Atomic Scattering to a Continuum Model
- Definitions- Scattering Length Density- Rayleigh-Gans Equation- Transmission
• Two - Phase Systems- Babinet’s Principle- Scattering Invariant- Porod Limit
• Multi-Phase Systems- Approaches to Determining Structures
Outline
Constructive interference from structures in the direction of q
Diffraction length scale
Scattering is at small angles - non-zero but smaller than
classical diffraction angles
Small Angle Scattering
kf
ki
q•
2θ
q2πd ≈
A1000to60A6
dλ2θ ≈≈
°°≈ 5to0.32θ
• Previously defined atomic cross sections:
• Easier to think in terms of material properties rather than atomic properties
• Define a “Scattering Length Density”
or
is the volume containing the n atoms
“Macroscopic” vs Atomic
2absinccoh 4ππσσσσ =
)rrδ(b)rρ( ii −=
v
bn
ii∑
=ρ
V
• Can we really use scattering length densities?
• consider
• We can use material properties rather than atomic properties when doing small-angle scattering
What Length Scales Are Probed?
2År30Åv:OH 32 ≈≈
ρ = bii∑
v
r*
HO
H HO
H
r
ρ_
molecules100
0.1Åq10Årrq1 1**
≈
≤≈> −
• Then from:
• We can replace the sum over atoms with integral over the scattering length density
• Normalizing by sample volume and introducing scattering length density:
• “Rayleigh-Gans Equation”• Inhomogeneities in give rise to small angle scattering
• is the “macroscopic cross section”
What Length Scales Are Probed?
)rρ(
∫2
V
rqi rde)rρ(V1)q(
dΩdσ
VN)q(
dΩdΣ ⋅==
∫V
N
ii r)drρ(b →∑
2N
i
rqiieb
N1)q(
dΩdσ ∑ ⋅=
VσΣ =
• Different types of systems have a natural basis - and all are equivalent
• This is especially true if the scattering is from “countable” units:
Polymers monomer unitParticulates per particleProteins polypeptide subunits
• A statistical description may also be appropriate:
Non-Particulate correlation function
Rayleigh-Gans Equation
∫ ∑∑ −→N
jji
N
i
2
V)rrf(rd)rf(
γ(r)ρ(r) →
• normalize by scattering volume
• Two contributions to measured signal:
• Incoherent scattering is not q-dependent and contributes only to the noise level, while absorption reduces the overall signal
Scattering from a Macroscopic Sample
section cross aldifferenti )q(dΩdσ =
eunit volumper scattering )q(dΩdσ
VN)q(
dΩdΣ ==
dΩdΣ)q(
dΩdΣ)q(
dΩdΣ inccoh +=
( )quantitymeasuredIntensity"Scattered"
I(q))q(dΩdΣ =∝
Interaction of Neutron Beam with Macroscopic Sample
dx
φo φ
0 tchange in flux
incident flux= • • thickness
total cross
section
dxΣd ⋅⋅=− φφ
vσΣ =
absinccohtotal ΣΣΣΣ ++=
remember:
• Integrating over the sample thickness:
• Tempting to solve for optimal sample thickness:
t = 1/Σ or Topt = 0.37 • but in practice ...
• incoherent scattering background• multiple scattering
• Higher transmissions are desired
Interaction of Neutron Beam with Macroscopic Sample
∫t0 dxΣd
0
−=⌡⌠
φ
φ φφ tΣln
0⋅−=
φφ
tΣ
0eontransmissi ⋅−==
φφ
• Incompressible phases of scattering length density ρ1 and ρ2
General Two-Phase System
21 VVV +=
=
22
11
Vin ρVin ρ
ρ(r)
ρ1ρ2
Then from the Rayleigh-Gans equation:• break the total volume into two sub-volumes
• So at non-zero q-values:
General Two-Phase System
∫ ∫2
V2
rqi2
V1
rqi1
21
rdeρrdeρV1)q(
dΩdΣ ⋅⋅ +=
∫ ∫ ∫2
V1
rqi
V
rqi2
V1
rqi1
11
rderdeρrdeρV1)q(
dΩdΣ
−+= ⋅⋅⋅
( ) ∫2
V1
rqi221
1
rdeρρV1)q(
dΩdΣ ⋅−=
Material Properties+
Radiation Properties
Spatial Arrangement of Material
Two structures give the same scattering
*incoherent scattering may be different
• Contrast is relative• Loss of phase information is ? • Very important in multi-phase systems
- contrast matching / variation
Babinet’s Principle
dΣdΩ
(q ) ∝ ρ1 − ρ2( )2
ρ1 > ρ2
10% black / 90% white in each square • Scattered intensity for each would certainly be different
• For an incompressible, two-phase system:
•Domains can be in any arrangement Guinier &Fournet, pp. 75-81.
Scattering Invariant
( ) ( )23 ρ)rρ(2πqd)q(dΩdΣQ~ −=⌡
⌠=
( )2bwbb
2* ρρ)(12πQ4πQ~ −−=≡ φφ
• At large q:
• S/V = specific surface area of sample
Porod Scattering
VS)q(I(q)lim
Qπ
elargq
4* =⋅⋅
→
A B
SAV
>SbV
4qI(q) −∝
• “contrast” and “structure” terms can still be factored asfor 2-phase system
Multi-Phase Materials
dΣdΩ
(q ) → dΣdΩ
(q,ρi ,Sij )
• For ‘p’ different phases (0,p)
• Scattering is now a sum of several terms with possibly many unknowns (Sij’s)
*Higgins & Benoit pp. 121-2.
Multi-Phase Materials
( ) ( )( ) (q)Sρρρρ(q)Sρρ(q)dΩdΣ
ijji
0j0i
p
1iii
20i ∑∑
<=−−+−=
• Contrast Matching - reduce the number of phases ‘visible’
ρ solvent = ρ core ρ solvent = ρ shell(shell visible) (core visible)
• The two distinct two-phase systems can be easily understood
Solving Multi-Phase Structures
becomes…
or
• A set of scattering experiments can yield a set of equations - of known contrasts and unknown ‘partial structure
functions’• Sturhmann Analysis
Determine structure from Rg = F(contrast)
Solving Multi-Phase Structures
• General treatment of small-angle scattering:- Rayleigh-Gans equation- Scattering length density- Specialized for specific systems of interest
• Two-phase systems:- Relative scattering length density- Model independent results
• Multi-phase systems:- Advanced techniques- Control of contrast
Summary