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) −∝

*Glatter & Kratky pp. 30-1.

Porod Scattering

log I(q)

log (q)

q-4A

B

• “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