•Fundamentals of neutron scattering 100•Neutron diffraction 101•Nobel Prize in physics

Neutron Scattering 102:SANS and NR

Pre-requisites:

Grade based on attendance and participation

Paul Butler

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

SANS and NR measures interference patterns from structures in the direction of Q

SANS and NR assume elastic scattering

QR kRki

2R

i f

QS

ki

kS

incident beamwavevector |ki|=2/ scattered beam

wavevector |kS|=2/

2s

Neutron Reflectometry (NR) Reflection mode

Small Angle Neutron Scattering (SANS) Transmission mode

f = i = R

kR = ki+QR

QR =4 sinR / Perpendicular to surface

kS = ki+Qs

Qs=|Qs|=4 sins /

Small Angle Neutron Scattering (SANS)

|3-D Fourier Transform of scattering contrast|2

normalized to sample scattering volume

S

V S

S V

rdrQirQ

d

d S

23.exp

Slide Courtesy of William A. Hamilton

Reciprocity in diffraction:Fourier features at QS => size d ~ 2/QS

Intensity at smaller QS (angle) => larger structures

Measure: Scattered Intensity => Macroscopic cross section = (Scattered intensity(Q) / Incident intensity) T d

Macromolecular structures: polymers, micelles,complex fluids, precipitates,porous media, fractal structures

Specular Neutron Reflection

|1-D FT of depth derivative of scattering contrast|2 / QR4

2

4

2

exp4

z

RR

R dzziQdz

zd

QQR

Slide Courtesy of William A. Hamilton

At lower QR, R reaches its maximum R=1 i.e. total reflection

Similar to SANS but ...This is only an approximation valid at large QR

of an Optical transform - refraction happens

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

Measure: Reflection Coefficient = Specularly reflected intensity / Incident intensity

Specular Reflectivity vs. Scattering length density profiles

Critical edgeR=1 for QR<QC

QC=4()1/2

T

Bragg peak

a

QR=2/aQR=2/T

sld step Thin film Multilayer

Fourier features (as per SANS)Fresnel reflectivity

Slide Courtesy of William A. Hamilton

Thin filmInterference

fringes

What SANS tells us

1

P(Q) = form factor (shape)

Q

S(Q) = Structure factor (interactions or correlations)or Fourier transform of g(r)

)()()( QSQAPQd

d coh

)()( QASQ

d

d coh

small θ … how

Sizes of interest = “large scale structures” = 1 – 300 nm or more0.02 < Q ~ 2/d < 6

Q=4 sin /

Cold source spectrum 3-5< <20A

Intensity balance sample size with instrument length

Cold Source Brightness

1.00E+09

1.00E+10

1.00E+11

1.00E+12

1.00E+13

0 5 10 15 20

Wavelength (A)n

eu

tro

ns

/cm

^2

/A/s

ter/

se

cApproaches to small θ:• Small detector resolution/Small slit (sample?) size• Large collimation distance

Δθ

Sizes of interest = “large scale structures” = 1 – 300 nm or more

SANS Approach QS

ki

kS

SSD SDD≈

S1 ≈ 2 S2

Optimized for ~ ½ - ¾ inch diameter sample

2 θ

S1

3m – 16m 1m – 15m

Sizes of interest = “large scale structures” = 1 – 300 nm or more

NR Approach

θ?

? = Ls sinθ

QR kRkiPoint by point scan

? ~ 1mm for low Q

Ls

10-2

10-1

100

101

102

103

104

105

0 100 4 10 -4 8 10-4 1.2 10-3

emptyEwald + Bgdlatex

q (Å-1)

IBGD

= 0.025 s-1

IPeak

= 60,000 s-1

Sizes of interest = “large scale structures” = 1 – 300 nm or more

QS

ki

kS

Ultra Small Angle Approach – when SANS isn’t small enough

Point by point scan - again

Fundamental Rule: intensity OR resolution… but not both

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)

Stray neutronsand Electronic noise

Incident beam

aperture

air

sample

cell

• Contribution to detector counts

Sample Scattering

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

At large q:

S/V = specific surface are

10 % black90 % white

)()()( 2 QSQPVQd

dp

coh

Imeas = Φ A ε t R +Ibgd t

Rocking Curve

i fixed, 2f varying

Specular Scan

2f = 2I

f = i

i 2f

Background Scan

f ≠ I

•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:SANSImeas(i) = Φ t A ε(i) ΔΩ Tc+s[(dΣ/dΩ)s(i) ds + (dΣ/dΩ)c(i) dc] +Ibgd t

NRImeas = Φ A ε t R +Ibgd t

Summary

)()()( 2 QSQPVQd

dp

coh

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)

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?

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?)

QR

kRki

D