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•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