1. Neutron Scattering 102: SANS and NR
Paul Butler
Pre-requisites:
Fundamentals of neutron scattering 100
Neutron diffraction 101
Nobel Prize in physics
Grade based on attendance and participation

2. Sizes of interest = “large scale structures” = 1 – 300 nm or more
Mesoporous structures
Biological structures (membranes, vesicles, proteins in solution)
Polymers
Colloids and surfactants
Magnetic films and nanoparticles
Voids and Precipitates

3. QR ki kR kS QS ki 2R SANS and NR assume elastic scattering
SANS and NR measures interference patterns from structures in the direction of Q
f = i = R
kR = ki+QR
QR =4 sinR / 
Perpendicular to surface
QR kR ki
2R i f
Neutron Reflectometry (NR)
Reflection mode QS ki kS
incident beam
wavevector |ki|=2/
scattered beam
wavevector |kS|=2/
2s
kS = ki+Qs
Qs=|Qs|=4 sins / 
Small Angle Neutron Scattering (SANS)
Transmission mode

4. Small Angle Neutron Scattering (SANS)
Macromolecular structures: polymers, micelles,complex fluids, precipitates,porous media, fractal structures
Measure: Scattered Intensity => Macroscopic cross section
= (Scattered intensity(Q) / Incident intensity) T d
|3-D Fourier Transform of scattering contrast|2
normalized to sample scattering volume
Reciprocity in diffraction:
Fourier features at QS => size d ~ 2/QS
Intensity at smaller QS (angle) => larger structures
Slide Courtesy of William A. Hamilton

5. Specular Neutron Reflection Measure: Reflection Coefficient
= Specularly reflected intensity / Incident intensity
Layered structures or correlations relative to a flat interface:
Polymeric, semiconductor and metallic films and multilayers, adsorbed surface structures and complex fluid correlations at solid or free surfaces
|1-D FT of depth derivative of scattering contrast|2 / QR4
Similar to SANS but ...
This is only an approximation valid at large QR
of an Optical transform - refraction happens
At lower QR, R reaches its maximum R=1 i.e. total reflection
Slide Courtesy of William A. Hamilton

6. Specular Reflectivity vs. Scattering length density profiles T a
sld step  
Thin film
Multilayer
QR=2/T
QR=2/a
Thin film
Interference fringes
Critical edge
R=1 for QR<QC
QC=4()1/2
Bragg peak
Fourier features (as per SANS)
Fresnel reflectivity
Slide Courtesy of William A. Hamilton

7. What SANS tells us
S(Q) = Structure factor (interactions or correlations)
or Fourier transform of g(r) 1
P(Q) = form factor (shape) Q

8. Sizes of interest = “large scale structures” = 1 – 300 nm or more
0.02 < Q ~ 2/d < 6
Q=4 sin / 
Cold source spectrum 
3-5<  <20A
 small θ … how
Approaches to small θ:
Small detector resolution/Small slit (sample?) size
Large collimation distance
Intensity  balance sample size with instrument length

9. Optimized for ~ ½ - ¾ inch diameter sample
Sizes of interest = “large scale structures” = 1 – 300 nm or more QS ki kS
SANS Approach
2 θ
S ≈ S2
DETECTOR S1 Δθ
3m – 16m
1m – 15m
SSD ≈
SDD
Optimized for ~ ½ - ¾ inch diameter sample

10. Sizes of interest = “large scale structures” = 1 – 300 nm or more
NR Approach
QR ki kR
Point by point scan Ls θ ?
? = Ls sinθ
? ~ 1mm for low Q

11. Point by point scan - again
Sizes of interest = “large scale structures” = 1 – 300 nm or more
Ultra Small Angle Approach – when SANS isn’t small enough QS ki kS
Point by point scan - again  
Fundamental Rule: intensity OR resolution
… but not both

12. Imeas(i) = Φ t A ε(i) ΔΩ Tc+s[(dΣ/dΩ)s(i) ds + (dΣ/dΩ)c(i) dc] +Ibgd t
Sample Scattering
Contribution to detector counts
1) Scattering from sample
2) Scattering from other than sample (neutrons still go through sample)
3) Stray neutrons and electronic noise (neutrons don’t go through sample)
aperture
sample
Incident beam
air
cell
Stray neutrons
and Electronic noise
Imeas(i) = Φ t A ε(i) ΔΩ Tc+s[(dΣ/dΩ)s(i) ds + (dΣ/dΩ)c(i) dc] +Ibgd t

13. SANS Basic Concepts 10 % black At large q: 90 % white
S/V = specific surface are

14. Imeas = Φ A ε t R +Ibgd t i 2f Rocking Curve i fixed, 2f varying
Specular Scan
2f = 2I
f = i
Background Scan
f ≠ I
Imeas = Φ A ε t R +Ibgd t

15. Summary SANS and NR measure structures in the direction of Q only
SANS and NR assume elastic scattering
SANS is a transmission technique that measures the average structures in the volume probed
NR is a reflection technique that measures the z (depth) density profile of structures strongly correlated to the reflection interface
Thinking aids:
SANS
Imeas(i) = Φ t A ε(i) ΔΩ Tc+s[(dΣ/dΩ)s(i) ds + (dΣ/dΩ)c(i) dc] +Ibgd t NR
Imeas = Φ A ε t R +Ibgd t

16. When measuring a gold layer on a Silicon substrate for example, many reflectometers can go to Q > 0.4 Å-1 and reflectivities of nearly However most films measured at the solid solution interface only get to 10-5 and a Qmax of ~ 0.25Å-1 Why might this be and what might be done about it. (hint: think of sources of background)
SANS is a transmission mode measurement, so with an infinitely thick sample the transmission will be zero and thus no scattering can be measured. If the sample is infinitely thin, there is nothing to scatter from…. So what thickness is best? (hint: look at the Imeas equation)
For a strong scatterer, a large fraction of the beam is coherently scattered. This is good for signal but how might it be a problem? (hint: think of the scattering from the back or downstream side of the sample)

17. USANS gets to very small angle
USANS gets to very small angle. However SANS is a long instrument in order to reach small angles. Why not make the instrument longer?
(Hint: particle or wave?)
Given the SANS pattern on the right, how can know what Q to associate with each pixel? (hint use geometry and the definition for Q)
NR and SANS measure structures in the direction of Q. Given the NR Q is in the z direction, can NR be used to measure the average diameter of the spherically symmetric object floating randomly below the interface?
QR kR ki D