Generalized Indirect Fourier Transformation (GIFT)

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Generalized Indirect Fourier Transformation (GIFT)
(see J. Brunner-Popela O.Glatter, J. Appl.
Cryst. (1997) 30, 431-442. Small-angle scattering
of interacting particles. I. Basic principles of
a global evaluation method) Non-dilute
systems no longer just solution of linear
weighted least-squares problem intraparticle
interparticle scattering must be
considered scattering intensity written as
product of particle form factor P(q) structure
factor S(q) leads to a highly nonlinear
problem
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Generalized Indirect Fourier Transformation (GIFT)
(see J. Brunner-Popela O.Glatter, J. Appl.
Cryst. (1997) 30, 431-442. Small-angle scattering
of interacting particles. I. Basic principles of
a global evaluation method) Non-dilute
systems generalized version of the indirect
Fourier transformation method - possible to
determine form factor structure factor
simultaneously no models for form
factor structure factor parameterized w/ up to
four parameters for given interaction model
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems For homogeneous isotropic
dispersion of spherical particles also
possible for non-spherical systems - structure
factor replaced by so-called effective
structure factor
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems For homogeneous isotropic
dispersion of spherical particles also
possible for non-spherical systems - structure
factor replaced by so-called effective
structure factor A major effect of S(q) is
deviation from ideal particle scattering curve
at low q
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Vector d contains
the coefficients dk (k 1-4) determining the
structure factor for the particles volume
fraction size (radius) polydispersity
parameter particle charge
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Then
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Then Accounting for
smearing
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Determine c? and dk by
usual weighted least squares procedure
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Determine c? s and dk s
by usual weighted least squares
procedure Complex problem, so separate into
2 parts. Use a fixed d to 1st get c? s
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Determine c? s and dk s
by usual weighted least squares
procedure Complex problem, so separate into
2 parts. Use a fixed d to 1st get c? s then
use fixed c? s to get dk s then iterate
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Simulation tests simulate
P(q), S(q,d) smear add noise get I(q)
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Simulation tests simulate
P(q), S(q,d) smear add noise get
I(q) determine initial values for dk s then
get c? s from
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Simulation tests simulate
P(q), S(q,d) smear add noise get
I(q) 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
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems 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)
Non-dilute systems Consider case of
monodispersed hard spheres w/ no charge (3 dk
s) Effect of volume fraction ?
? 0.35
? 0.15
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Consider case of
monodispersed hard spheres w/ no charge (3 dk
s) Effect of radius RHS
RHS 6 nm
RHS 14 nm
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Consider case of hard spheres
w/ no charge (3 dk s) Effect of polydispersity
?
? 0
? 0.6
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Simulated data for
homogeneous spheres (? 0.15, RHS 10 nm, ?
0.4)
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Simulated data for
homogeneous 11 nm x 21 nm cylinders (? 0.15,
RHS 12 nm, ? 0.4)
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Simulated data for
non-homogeneous spheres (? 0.285, RHS 10
nm, ? 0.3)
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Simulated data for
non-homogeneous spheres (? 0.285, RHS 10
nm, ? 0.3)
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Simulated data for
non-homogeneous spheres (? 0.285, RHS 10
nm, ? 0.3)
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Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems Simulated data for
non-homogeneous 11 nm x 29 nm cylinders (?
0.15, RHS 12 nm, ? 0.4)
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Generalized Indirect Fourier Transformation (GIFT)
Comments Min. amt of info system required No
models - only require hard spheres type
interaction polydispersity expressed by an
averaged structure factor No assumptions as to
particle shape, size, distrib., or internal
structure Not completely valid (as of 1997) for
highly dense systems, true polydispersed
systems, or highly non-spherical
particles