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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
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 transformationmethod - possible to determine form factor &structure factor simultaneously
no models for form factor
structure factor parameterized w/ up to four parameters forgiven interaction model
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
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
Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems
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 fractionsize (radius)polydispersity parameterparticle charge
Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems
Then
Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems
Then
Accounting for smearing
Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems
Determine c and dk by usual weighted least squares procedure
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 1stget c s
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 1stget c s then use fixed c s to get dk s
then iterate
Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems
Simulation tests:
simulate P(q), S(q,d)smearadd noiseget I(q)
Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems
Simulation tests:
simulate P(q), S(q,d)smearadd noiseget I(q)
determine initial values for dk sthen get c s from
Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems
Simulation tests:
simulate P(q), S(q,d)smearadd noiseget I(q)
determine initial values for dk sthen get c s from
determine dk s from above
iterate until final c s and dk s obtained
Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems
determine initial values for dk sthen 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
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
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
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
Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems
Simulated data for homogeneous spheres ( = 0.15, RHS = 10 nm, = 0.4)
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)
Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems
Simulated data for non-homogeneous spheres ( = 0.285, RHS = 10 nm, = 0.3)
Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems
Simulated data for non-homogeneous spheres ( = 0.285, RHS = 10 nm, = 0.3)
Generalized Indirect Fourier Transformation (GIFT)
Non-dilute systems
Simulated data for non-homogeneous spheres ( = 0.285, RHS = 10 nm, = 0.3)
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)
Generalized Indirect Fourier Transformation (GIFT)
Comments
Min. amt of info ~ system requiredNo models - only require hard spheres type interaction & polydispersity
expressed by an averaged structure factorNo assumptions as to particle shape, size, distrib., or internal structureNot completely valid (as of 1997) for highly dense systems, true polydispersed
systems, or highly non-spherical particles
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