Neutron Scattering 102: SANS and NR

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 i f 2R kS QS 2s ki incident beam wavevector|ki|=2/ scattered beam wavevector|kS|=2/ SANS andNRassumeelastic scattering SANSand 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 Neutron Reflectometry (NR) Reflection mode 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. T  QR=2/T QR=2/a Specular Reflectivity vs. Scattering length density profiles a sld step  Thin film Multilayer 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. kS QS ki Sizes of interest = “large scale structures” = 1 – 300 nm or more SANS Approach 2 θ S1≈ 2 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. kS QS ki Sizes of interest = “large scale structures” = 1 – 300 nm or more Ultra Small Angle Approach – when SANS isn’t small enough Point by point scan - again   Fundamental Rule: intensity OR resolution … but not both

12. 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 Stray neutrons and Electronic noise cell Imeas(i) = ΦtA ε(i) ΔΩ Tc+s[(dΣ/dΩ)s(i) ds + (dΣ/dΩ)c(i) dc] +Ibgd t

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

14. Specular Scan 2f = 2I f = i Background Scan f ≠ I Rocking Curve i fixed, 2fvarying i 2f Imeas = Φ AεtR+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) = ΦtA ε(i) ΔΩ Tc+s[(dΣ/dΩ)s(i) ds + (dΣ/dΩ)c(i) dc] +Ibgd t • NR • Imeas = Φ AεtR+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 10-8. 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. kR ki QR D 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?