Generalized Indirect Fourier Transformation (GIFT)

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Generalized Indirect Fourier Transformation (GIFT)
(see B. Weyerich, J. Brunner-Popela O. Glatter,
J. Appl. Cryst. (1999) 32, 197-209. Small-angle
scattering of interacting particles. II.
Generalized indirect Fourier transformation under
consideration of the effective structure factor
for polydisperse systems) Previous
GIFT actually assumed a simplistic model for
structure factor the averaged structure
factor
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Generalized Indirect Fourier Transformation (GIFT)
(see B. Weyerich, J. Brunner-Popela O. Glatter,
J. Appl. Cryst. (1999) 32, 197-209. Small-angle
scattering of interacting particles. II.
Generalized indirect Fourier transformation under
consideration of the effective structure factor
for polydisperse systems) Previous
GIFT actually assumed a simplistic model for
structure factor the averaged structure
factor for monodisperse particles Now consider
another model - the "effective structure factor"
for hard spheres with a better treatment of
polydispersity
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Generalized Indirect Fourier Transformation (GIFT)
For monodisperse, homogeneous, isotropic
dispersion of spherical particles
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Generalized Indirect Fourier Transformation (GIFT)
For monodisperse, homogeneous, isotropic
dispersion of spherical particles Suppose
mixture of m components - the components
here are different-sized homogeneous
spheres Each sphere has a unique form
amplitude ƒ? at q 0 normalized form amplitude
B? so that (Blum Stell, 1979)
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Generalized Indirect Fourier Transformation (GIFT)
For monodisperse, homogeneous, isotropic
dispersion of spherical particles Suppose
mixture of m components - the components
here are different-sized homogeneous
spheres Each sphere has a unique form
amplitude ƒ? at q 0 normalized form amplitude
B? For this system (Blum Stell,
1979) structure factor now for
inter- action of different-sized spheres
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Generalized Indirect Fourier Transformation (GIFT)
For monodisperse, homogeneous, isotropic
dispersion of spherical particles Suppose
mixture of m components Then define an
averaged form factor x?? molar fraction
of ? so that
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Generalized Indirect Fourier Transformation (GIFT)
For monodisperse, homogeneous, isotropic
dispersion of spherical particles Suppose
mixture of m components Then define an
averaged form factor x?? molar fraction
of ? so that
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Generalized Indirect Fourier Transformation (GIFT)
Suppose mixture of m components Then
define an averaged form factor so
that Thus Note that Seff(q) depends on
both the particle interactions the particle
form amplitudes
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Generalized Indirect Fourier Transformation (GIFT)
Note that Seff(q) depends on both the
particle interactions the particle form
amplitudes Previously, averaged structure
factor used for Seff(q) (weighted addition of
partial structure factors S?(q) for a
monodisperse system of particles ?, each having a
different radius)
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Generalized Indirect Fourier Transformation (GIFT)
Other models a. local monodisperse
approximation accounts for dependence on f,
B, but not correlations betwn different-sized
particles
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Generalized Indirect Fourier Transformation (GIFT)
Other models a. local monodisperse
approximation b. decoupling
approximation R(q) accounts for the
different scattering properties of the
particles Monodisperse S(q) corrected by
'incoherent scattering' term R(q)
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Generalized Indirect Fourier Transformation (GIFT)
Other models a. local monodisperse
approximation b. decoupling
approximation To calculate S?(q), use mean
spherical approxn (Percus Yevick,1958)
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Generalized Indirect Fourier Transformation (GIFT)
Simulation tests simulate P(q),
S(q) smear add noise get I(q)
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Generalized Indirect Fourier Transformation (GIFT)
Simulation tests simulate P(q),
S(q) smear add noise get I(q) determine
initial values for dk s for S(q) then get c? s
from
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Generalized Indirect Fourier Transformation (GIFT)
Simulation tests simulate P(q),
S(q) smear add noise get I(q) determine
initial values for dk s for S(q) then get c? s
from determine dk s from above iterate
until final c? s and dk s obtained
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Generalized Indirect Fourier Transformation (GIFT)
Tests determine initial values for dk s then
get c? s from determine dk s from above
iterate until final c? s and dk s
obtained finally use c? s to get pddf
pA(r) dk s directly give info on vol.
fract., polydispersity distrib., hard sphere
radius, charge
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Generalized Indirect Fourier Transformation (GIFT)
Compare Seff(q) for polydispersed system of
homogeneous spheres w/ ? 0.3, ?? 0.3
P-Y Seff
Slma
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Generalized Indirect Fourier Transformation (GIFT)
Compare Seff(q) Save (q) for polydispersed
system of homogeneous spheres form factor
assumed for homogeneous sphere w/ R 10 nm

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Generalized Indirect Fourier Transformation (GIFT)
Core/shell system
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Generalized Indirect Fourier Transformation (GIFT)
Core/shell system note strong
dependence of Seff(q) on polydispersity
at low q
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Generalized Indirect Fourier Transformation (GIFT)
Core/shell system
P-Y Seff
Slma