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

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

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

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

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

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

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

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

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 S1 ≈ 2 S2 DETECTOR S1 Δθ 3m – 16m 1m – 15m SSD ≈ SDD Optimized for ~ ½ - ¾ inch diameter sample

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

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

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

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

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

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

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)

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