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
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• 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)
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
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,