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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

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neutron scattering 102 sans and nr
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

slide2

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
slide3

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

slide4

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

slide5

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

slide6

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

slide7

What SANS tells us

S(Q) = Structure factor (interactions or correlations)

or Fourier transform of g(r)

1

P(Q) = form factor (shape)

Q

slide8

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

slide9

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

slide10

?

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

slide11

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

slide12

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

sans basic concepts
SANS Basic Concepts

At large q:

10 % black

90 % white

S/V = specific surface are

slide14

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

slide15

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
slide16

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)

slide17

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?