Small Angle Neutron Scattering
• Eric W. Kaler, University of Delaware
• Outline– Scattering events, flux– Scattering vector– Interference terms– Autocorrelation function– Single particle scattering– Concentrated systems– Nonparticulate scattering
Small Angle Neutron Scattering• Measures (in the ideal world…)
• Particle Size• Particle Shape• Polydispersity• Interparticle Interactions• Internal Structure
• Model-free Parameters• Radius of gyration – Rg• Specific surface – S/V
• Compressibility: ⇒ molecular weight
• Much information as part of an integrated approach involving many techniques
1−
⎟⎟⎠
⎞⎜⎜⎝
⎛∂Π∂ρ
References
• Vol. 21, part 6, Journal of Applied Crystallography, 1988.
• Chen, S.-H. Ann. Rev. Phys. Chem. 37, 351-399 (1986).
• Hayter, J.B. (1985)in Physics of Amphiphiles: Micelles, Vesicles, and Microemulsions, edited by V. Degiorgio, pp. 59-93.
• Roe, R.-J. (2000) Methods of X-ray and Neutron Scattering in Polymer Science.
Different Radiations
• Light (refractive index or density differences)– laboratory scale, convenient– limited length scales, control of scattering events (contrast)– dynamic measurements (diffusion)
• X-rays (small angle) (electron density differences)– laboratory or national facilities– opaque materials– limited contrast control
• Neutrons (small angle) (atomic properties)– national facilities– great contrast control
Neutrons• Sources
– nuclear reactor• US: NIST
– spallation sources (high energy protons impact a heavy metal target)
• US: Spallation Neutron Source (SNS)– 1.4 billion dollars, complete 2006
• Both cases produce high energy neutrons that must be ‘thermalized’ for materials science studies
SNS Overview
www.sns.gov
Maxwell Distribution
velocity distribution: ⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛= kTmvv
kTmvf /
21exp
24)( 22
2/3
ππ
Cold Neutrons: D2O at 25KThermal Neutrons: D2O at 300KHot Neutrons: Graphite at 2000K
mvh
=λ
T = 25Kv = 642 m/sE = 2.16 meVλ = 6.2 A
Flux, cross-section, intensity• Flux (plane wave): number per unit area per second
– for a wave of amplitude A, the flux • Flux (spherical wave): number per unit solid angle per
second
*2 AAAJ ==
Ω==
dd
JJ
o
σ beamincident theofflux
secondper direction given ain angle solidunit into scattered particles ofnumber
differentialscatteringcross-section
scattering angle = 2θ
2θdΩincident flux J0
scattered flux J
= intensity (I)
Bragg Condition for Interference
θ
A
B
C
Dd
θsindCDBC 2=+
ACBCsin =θ
ACDCsin =θ
θλ sindm 2=
When extra distance is equal to one wavelength
Dealing with Colloidal Dimensions
λθ
θλ
sin21
sin2
=
=
d
or
dm
Recall that interference between two particles is a function of the scattering angle and the separation between scattering centers
θ d
θ2
so the size explored varies inversely with the scattering angle
For d = 100 nm, λ = 1nm, sinθ = 0.005, so θ ≈ 0.005
Constructive interference
Destructive interference
Example of Interference
Constructive and destructive interference can lead to more (or less) intensity
Another Example of Interference
Constructive and destructive interference from path length differences
Light waves only interfere if they are polarized in the same direction.
Interference Calculation
( )okkqrrr
−=
P (j)Q
θO
R (i)
ji rr rr−
okr
kr
Consider a plane wave
)/(2exp(),( λπ xvtiAtxA −−=
with direction given bythe wave vector ko
Then the phase difference between the two waves scattered
from O and P is rqrkrkORQP o ⋅−=⋅−⋅
=−
==Δλ
πλ
πλπδϕ
)(2)(22
scattering vector
vector unit a is s where s
λ2πk ooo =
The scattering vector
( )
λθπ sin4
==
−=
kkq o
r
rrr
okr
kr
−
θqkko =−
rr
q lies in the plane of the detector
Notation: q, k, h, s =q/2π
The Value of the Scattering Vector Corresponds to a Distance in Real Space
Characteristic distance, d, that is measured in the experiment
1054.0~22
sin4 −=⎟⎠⎞
⎜⎝⎛= nm
dnq πθ
λπr
refractive index(~ 1.33 for water)
Wavelength(Argon = 514 nm)
scattering angle(often 90o)
nm~d 116
LightScattering
Small AngleScattering
1054.0~22
sin4 −=⎟⎠⎞
⎜⎝⎛= nm
dnq πθ
λπr
refractive index = 1
Wavelength1 nm
scattering angle(1 mrad)
nm~d 116
Comparison of light and small-angle x-ray or neutron scattering
0 0.003 0.2
q (Ǻ-1)
ISmall angle scattering
Light scattering
Small Angles… Big Machines
http://www.ncnr.nist.gov/instruments/ng3sans/ng3_sans_photos.html
Small Angle Scattering Instrument (NG-7) at NIST, Gaithersburg, MD
SANS Data ReductionNIST examples
Two Dimensional Data Reduced to I(q)
Interference continued• Now write the (spherical) scattered wave from particle 1 (at O)
• And the spherical scattered wave from particle 2 (at P)
• The combined wave on the detector is A = A1 + A2
• And the flux is
))/(2exp(),(1 λπ xvtibAtxA o −−=scattering length
incident amplitude
)exp())/(2exp()exp(),(),( 12 riqxvtibAitxAtxA o ⋅−−−=Δ= λπφ
))exp(1))(/(2exp(),( riqxvtibAtxA o ⋅−+−−= λπ
))exp(1))(exp(1(),(),( 22* riqriqbAtxAtxAJ o ⋅+⋅−+==
which only depends on q·r
Interference continued
• For N scatterers,
• and for a distribution of scatterers
• where n(r)dr is the number of scatterers in a volume element and V is the sample volume.
)exp()(1
j
N
jo riqbAqA ∑
=
⋅−=
rdriqrnbAqAV
o3)exp()()( ∫ ⋅−=
So What is Special About Neutrons?
• Neutrons have spin ½• Neutrons are scattered from atomic nuclei,
and the scattering event depends on the nuclear spin.
• There are coherent and incoherent scattering lengths tabulated for elements and isotopes– coherent – information about structure– incoherent – arises from fluctuations in scattering
lengths due to nonzero spins of isotopes and has no structural information
Neutron cross-sections• Hydrogen is special. Spin =1/2, with different spin up
and spin down scattering, gives rise to a very large incoherent scattering (this is bad for structure measurements, but good for dynamics)
• Deuterium is spin 1, with much lower incoherent scattering
Element bcoh(10-12cm)1H -0.3742D 0.667C 0.665O 0.580
For a molecule, thescattering length density
SLD=Σbi/molecular volume
H/D substitution changes the scattering power and givescontrol of n(r): this is called contrast variation.
Autocorrelation Function
• Setting Ao = 1, defining the scattering lengthdensity ρ(r) = Σn(r) b then
2
32
2
32
3
)exp()()()(
)exp()()()(
)exp()()(
rdriqrqAqI
rdriqrqAqI
rdriqrqA
V
V
V
∫
∫
∫
⋅−==
⋅−==
⋅−=
ρ
ρ
ρ
and
weak scattering(kinematic theory)
really anensemble average…
• With some calculus…
[ ][ ][ ]
∫
∫
∫ ∫
∫∫
+≡
=
+=
=′′=
=
−
−
−′−
duruurpwhere
drerp
rdeduruu
dueuudeu
qAqAqI
iqr
iqr
iquuiq
)()()(
)(
)()(
)()(
)()()( *
ρρ
ρρ
ρρ
u'-ur set and
is the autocorrelation function of ρ(r) and isthe Fourier transform pair of I(q)
Data Analysis
∫
∫+≡
= −
duruurp
drerpqI iqr
)()()(
)()(
ρρ
To find ρ(r), either1. Inverse Fourier Transform2. Propose a model and fit the measured I(q)
Data Interpretation
aggregate structure0.01
2
46
0.1
2
46
1
2
46
10
Scat
teri
ng In
tens
ity in
1/c
m
0.0012 3 4 5 6
0.012 3 4 5 6
0.12 3 4 5 6
1
Scattering Vector q in 1/Å
I = f(q)
approximate picture ofaggregate structure
experiment
model
interpretation
0 10 20 30 400.0
5.0x10-5
1.0x10-4
1.5x10-4
2.0x10-4
2.5x10-4
p c(r)
r
Generalized IndirectFourier Transform
(GIFT)
model
distance pair distribution functionp(r) = Σcνϕν(r)
Glatter, O. J. Appl. Cryst. 1977, 10, 415-421; 1980, 30, 431-442
Direct Model
experiments
assumption # of splines
distance pair distribution function
IFT
IFT
Indirect Fourier Transform
Linear Com
binationMethod of Global Indirect Fourier Transform
DILUTE LIMIT: Scattering from ParticlesIntraparticle Interference
Scattering from larger particles can constructively/destructively interfere, depending on size (relative to the size of the object) and shape of the particles.
Size (how big is big?)• Scattering vector, q, which gives the length probed• Introduce dimensionless quantity, ‘qR’, that indicates how big the particles are relative to the wavelength.
Shape• Introduce the Form Factor, P(q), the define the role of particle shape in the scattering profiles• P(q) for Spheres, leading to Guinier Plots• P(q) for vesicles, which are different than spheres• P(q) for Gaussian Coils/Polymers, leading to Zimm Plots
Generate constructive and destructive interference which is related to FORM
A some angle, the effect depends on the wavelength of the light, size of the aggregate and the shape of the aggregate.
each point scatters
Intraparticle Interference Arises from Scattering from the Particle
Introduce a Dimensionless Quantity to Answer the Question ‘How Long is Long?’
π>>qR
small-q limit
π≈qRπ<<qR
large-q limit
collective properties individual properties
Intraparticle Form Factor, P(q) is an Integral Over the Structure
∫ •−=v
riq rderqA 3)()( ρ
radial density of the particle
phase difference for two scatterersin the volume (as with definition of q)
Integral over the volume of the sample
Each shape is different, so each integral and each form factor will be different
)()(1)( 2 qPnqANV
qI pp ==
P(q) is the particle form factor
Form Factors for Spheres
Perry, R.L., and Speck, E.P. "Geometric Factors for Thermal Radiation Exchange Between Cows and Their Surroundings", American Society of Agricultural Engineers Paper #59-323.
For evaluating thermal radiant exchange between a cow and her surroundings, the cow can be represented by an equivalent sphere. The height of the equivalent sphere above the floor is 2/3 of the height at the withers. The origin of the sphere is about 1/4 of the withers-to-pin-bone length back of the withers. The sphere size differs for floor and ceiling, side walls, and front and back walls. For the model surveyed, the radius of the equivalent sphere is 2.13 feet for exchange with floor and ceiling, 2.38 feet for side walls, and 2.02 feet for the front and back walls. These values are 1.8, 2.08, and 1.78 times the heart girth. An equation in spherical coordinates is given for the variation of the size of the equivalent sphere with the angle of view measured from the vertical and transverse axes.
Form Factor for a Cow
The shape factor for exchange with an adjacent cow in a stanchion spacing of 3'8" was found to be 0.1.
θ
Form Factor for Sphere
( )2
3 ))cos()(sin()(
3⎟⎟⎠
⎞⎜⎜⎝
⎛−= qRqRqR
qRqRP
Integrate the scattering over the entire sphere, which gives an
analytical solution to the intra-particle form factor.
θ
Form Factor for Sphere2
32 )()()( ∫ •−==v
riq rderqAqP ρ
solid sphere of radius R, ∆ρ=ρ- ρsolvent
drriqr
qri
drriqr
ee
drdxre
ddrreqA
qrrq
dddrreqA
R
R iqriqr
Riqrx
Riqr
Rriq
2
0
2
0
2
0
1
1
2
0 0
cos
2
0 0
2
0
sin2)2(
)2(
)2(
sin)2()(
cos
sin)(
∫
∫
∫ ∫
∫ ∫
∫ ∫ ∫
Δ=
−Δ=
Δ=
Δ=
=⋅
Δ=
−−
−
−
•−
πρ
πρ
πρ
θθπρ
θ
φθθρ
πθ
π π
2
3222
3
333
22
0
cossin3)()()(
cossin3
)(cos
)(sin)4(
cossin4
sin4)(
⎥⎦⎤
⎢⎣⎡ −
Δ==
⎥⎦⎤
⎢⎣⎡ −
Δ=
⎥⎦
⎤⎢⎣
⎡−Δ=
⎥⎦
⎤⎢⎣
⎡−Δ=
Δ= ∫
xxxxVqAqP
xxxxV
qRqRqR
qRqRR
qqRqR
qqR
q
drqrrq
qA
p
p
R
ρ
ρ
πρ
πρ
πρ
Interference Plots for Spheres
q (A-1)
0.000 0.001 0.002 0.003 0.004 0.005 0.006
P(q)
0.0
0.2
0.4
0.6
0.8
1.0 30 nm
96 nm
500 nm
30o (q ~ 8 x 10-4 A-1)150o (q ~ 3.1 x 10-3 A-1)
Shape of the Form FactorInterference Plots for Spheres
q (A-1)
0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
P(q)
1e-10
1e-9
1e-8
1e-7
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
1e+2
30 nm
96 nm
500 nm
30o (q ~ 8 x 10-3 A-1)
150o (q ~ 3.1 x 10-2 A-1)qr ~ 4.49
The sizes denote the diameter of the particles; the red lines denote the q values accessible with typical light scattering measurements
Sphere Form Factor
• 6 nm monodisperse sphere
Guinier Expression
3
22qR
o
g
eII−
=
...!
xxe x −+−≈−2
12
Note that an exponential can be expanded as a power series
31...1)(
2242 qR
bqaqqPII g
o
−≈++≈=
Back in the day… intensities were weak, so special carewas taken at the low-q region
so this suggested that in general
where Rg is the radius of gyration
(Guinier radius)
General Feature - Guinier Region (qa < π)
3
22qR
oo
g
eI)q(PI)q(I−
≈=
3
22qRIlnIln go −=
q2
ln(I
)Then, taking the natural logarithm of the expression
3
22qRg−
Onset of the angle dependence of the scattering
The plotting the ln I versus q2, leads to a plot with the slope proportional to the square of the scattering vector
Guinier Plot
Guinier continued
• In general, for monodisperse objects
• Example- solid sphere
• Aside – for polydisperse spheres measure <Rg2>z
∫∫
−
−=
rdr
rdrrR
s
sg 3
322
))((
))((
ρρ
ρρ
22
222
53 R
drr
drrrRg ==
∫∫ or Rg = 0.77 R
How Good Are Guinier Approximations?Guinier Approximation for 30 nm Beads
q (A-1)
0.000 0.001 0.002 0.003 0.004 0.005
P(q)
0.0
0.2
0.4
0.6
0.8
1.0
• Guinier Approximations work well provide ‘qa’ is small (black- full expression of P(q); blue- Guinier Approximation)
• As particles get larger, the angles must be far smaller• Limit ~ 100 nm for LS measurements, using smaller angles
Guinier Approximation for 96 nm Beads
q (A-1)
0.000 0.001 0.002 0.003 0.004 0.005
P(q)
0.0
0.2
0.4
0.6
0.8
1.0
Anisotropic Scatterers
• Rods or disks may not always be isotropic– Above analysis is for I(q) = I(q)
• Alignment may give additional information
Porod Region (qa >> π)
q ~ a( )
2
3 ))cos()(sin()(
3⎟⎟⎠
⎞⎜⎜⎝
⎛−= qRqRqR
qRqRP
Recall that…
In the limit that ‘qR’ is large
( )( ) ( )
( ) 44
22
2
2
3
1)(cos1)(
)cos(3 −≈≈⎟⎟⎠
⎞⎜⎜⎝
⎛≈⎟⎟
⎠
⎞⎜⎜⎝
⎛ −≈ qR
qRqR
qRqRqRqRqRP
‘qR’ dominates summation
P(q) is Dominated by q-4 Term
Porod Scattering for 50 nm Sphere
q (A-1)
0 5 10 15 20 25 30
P(q)
1e-191e-181e-171e-161e-151e-141e-131e-121e-111e-101e-91e-81e-71e-61e-51e-41e-31e-2
P(q)
Porod Envelop
Form Factor for Vesicles
Form Factor of Vesicles Versus Spheres
2))(()( qAqP =
))cos()(sin(3)( qRqRqRqR
qRA −=
2))q(F)q(F()q(P insideoutside −=
2
13
132
333
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎟⎠
⎞⎜⎜⎝
⎛
−= )qR(J
qRR)qR(J
qRR
RR)q(P io
i
i
o
o
io
( ) qR)qRcos(
qR)qRsin()qR(J −= 21
Form factor for a sphere is given as:
Form factor for a vesicle is outside sphere minus the inside spheres
Where J1(q) is the first order Bessel Function
Form Factor of Vesicles Versus SpheresScattering from Vesicles
q (A-1)
0.000 0.005 0.010 0.015 0.020
Y D
ata
0.0
0.2
0.4
0.6
0.8
1.0
150o (~ 3.1 x 10-3 A-1)30o (~ 8.4 x 10-4 A-1)
300 A500 A
700 A
1000 A
500 A (solid sphere)
Which looks very different than a sphere, for the same size
Contrast variation• Consider a core and shell morphology:
• and change the solvent (H/D) to match the SLD of the core and shell, separately
Contrast Variation for Composite Particle
Clean sphere scattering Clean shell scatteringgives core dimension gives shell dimension
There are Other Forms of P(q)Thin Rods: Length 2H; Diameter 2R; at low q
( )qH
eqP/Rq
2
422−=
32
22 HRRg +=
Disk: Thickness 2H; Diameter 2R
( ) 22
322
RqeqP
/Hq−=
32
22 HRRg +=
Fractal Region (qa ~ π)
q ~ aq ~ size of the
aggregate
• Small q ~ size of the individual particles• Large q ~ size of the individual aggregates
Random fractal objects produced by using the band-limited Weierstrass functions and employed in experiments. Assigned fractal dimension was D = (a) 1.2, (b) 1.5, and (c) 1.8.
The Shape of Different Fractal Particles
Fractal Region for Aerosol Aggregates
182
305000
3314 −≈⎟⎠⎞
⎜⎝⎛= msin
m.).(q μ
μπr
1302
1305000
3314 −≈⎟⎠⎞
⎜⎝⎛= msin
m.).(q μ
μπr
md μ1≈
m.d μ10≈
fdoo eI)q(P)q(I)q(I −
≈= qlndIln f−≈
Allowing Characterization Over Many Distances
Logarithm-logarithm plots result in slopes that relate to the different levels of structures
Scattering from Particulate Systems
)exp()(1
j
N
jo riqbAqA ∑
=
⋅−=Recall:2
)( ∑ ⋅=≈Ω i
riqi
iebqIddσ so
xi
Ri write ri = Ri + xi
ri
Scattering from Particulate Systems
2
1
2
)(
∑∑
∑
⋅
=
⋅
⋅
=
=≈Ω
celli
xiqij
N
i
Riq
i
riqi
jp
i
i
ebe
ebqIddσ so
sum over the number of cells
sum over the scatterers in each cell
Scattering from Particulate Systems
above! from
so
uniform) and (constant solvent the in
define particle, the in
cell each for factor' 'form a define now
i
i
i
i
)(
)())((0
))(()(
)(
)()(
)(
qA
qdrer
dredrerqA
r
xrbr
ebqA
riqs
particle
riq
cellis
riqs
cellii
s
jjij
celli
xiqiji
j
=
+−+=
+−=
=
−=
=
⋅
⋅⋅
⋅
∫
∫∫
∑
∑
δρρ
ρρρ
ρρ
δρ
Scattering from Particulate Systems
2
1
2
1
2
)(
)(
qAe
ebe
ebqIdd
i
N
i
Riq
celli
xiqij
N
i
Riq
i
riqi
p
i
jp
i
i
∑
∑∑
∑
=
⋅
⋅
=
⋅
⋅
=
=
=≈Ωσ so
particle shape, size, polydispersity
arrangement of particle centers
Scattering from Particulate Systems
2
2
1
)()(
)()(
qANqI
qAeqIdd
ip
i
N
i
Riqp
i
=
=≈Ω ∑
=
⋅
so
ed,uncorrelat are R the dilute'' are particles the when
so
i
σ
as before!
So, how do we find the Ri ‘s??
Scattering from Particulate Systems
)))(exp(11)((
))(exp()()()(
)()(
)()())(exp(1)(1
)()(
1 1
1 1
2
1 1
*
1
2
2
1
∑∑
∑∑
∑∑∑
∑
=≠=
=≠=
=≠==
=
⋅
−⋅+=
−⋅+=
=
−⋅+=
=≈Ω
p p
p p
p pp
p
i
N
i
N
ijj
jip
p
N
i
N
ijj
jip
i
N
i
N
ijj
jiji
N
ii
i
N
i
Riq
RRiqN
qPn
RRiqV
qPV
qPNqI
qPqA
qAqARRiqV
qAV
qAeqIdd
allspheres, semonodisper of case simpliest the in again,
so σ
Scattering from Particulate Systems
]sin)1)((41)[()(
))(exp(1
2
0
1 1
drrqr
qrrgnqPnqI
RRiqN
pp
N
i
N
ijj
jip
p p
∫
∑∑
∞
=≠=
−+=
−⋅
π
as equation working master the write finally can we sog(r), function ondistributi radial micthermodyna the to related is
Fundamental working equation for monodisperse sphericalparticles, with the term in brackets called thestructure factor, so
I(q) = npP(q)S(q)
Structure Factor
• For 5 nm hard spheres, 20% volume fraction
Scattering from Particulate Systems
So how do we get S(q)?
Various thermodynamic models relate g(r) (and thus S(q))to the interparticle potential
There are two questions:1. What is the nature of the potential?
Hard sphere?Electrostatic?Depletion?Steric?
2. What thermodynamic formalism do you use tocalculate g(r)?
Scattering from Particulate SystemsPotential Solution (closure) Comments
Hard Sphere Percus-Yevick ExcellentRogers-Young analytic,
can be extendedto polydisperse
Electrostatic Mean-Spherical MonodisperseApproximation
Square Well Sharma&Sharma (PY) Monodisperse
And many more… verified by computer simulations
Scattering from Particulate SystemsWhat about the real world…
polydisperse, nonspherical…
Various ‘decoupling approximations’ to deal with theissues of
These are best for repulsive potentials.
Data workup: http://www.ncnr.nist.gov/programs/sans/manuals/available_SANS.html
∑∑=
≠=
−⋅p pN
i
N
ijj
jiji qAqARRiqV 1 1
* )()())(exp(1
Non-Particulate Scattering
Using a free energy model derive correlation function for bicontinuous structures
)/rexp(d
r2sinr2
d)r( ξ−⎟⎠⎞
⎜⎝⎛ π
π=γ4
22
12
22
qcqca/Vc8
)q(I++
ξηπ=
3-D Correlation FunctionScattering Function
FourierTransform
d : repeat length of microemulsion(oil + water domain)
ξ : correlation length
Structure characterized by 2 parameters:
Example: Teubner-Strey model for bicontinuous microemulsions
Amphiphilicity Factor
I(q)
q
γ(r)
r
fa ~ 1
fa ~ 0
fa ~ -1
disorder
order
“good”
lamellar
22
1a ca4
cf =
1
10
100
0.001 0.01 0.1 1
1
10
100
0.001 0.01 0.1 1
1
10
100
1000
0.001 0.01 0.1 1-0.20.00.20.40.60.81.0
0 100 200 300 400
-0.20
0.20.40.60.8
1
0 100 200 300 400
-0.20.00.20.40.60.81.0
0 100 200 300 400
125
130
135
140
145
150
D (A
)
50
55
60
65
70
75
80
-1
-0.95
-0.9
-0.85
-0.8
-0.75
-0.7
0 10 20 30δ(wt %)
f a
Amphiphilicity Factor
fa
disordered
ordered
"good"
lamellar
1
0
-1
increasing order
Adding ionic surfactant to surfactant monolayer
