Generalized Indirect Fourier Transformation (GIFT)

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


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


Vector d contains the coefficients dk (k = 1-4) determining the structure factor for the particles

volume fractionsize (radius)polydispersity parameterparticle charge


Then


Then

Accounting for smearing


Determine c and dk by usual weighted least squares procedure


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


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


Simulation tests:

simulate P(q), S(q,d)smearadd noiseget I(q)


Simulation tests:


determine initial values for dk sthen get c s from


Simulation tests:



determine dk s from above

iterate until final c s and dk s obtained



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


Consider case of monodispersed hard spheres w/ no charge (3 dk s)

Effect of volume fraction

= 0.35

= 0.15


Consider case of monodispersed hard spheres w/ no charge (3 dk s)

Effect of radius RHS

RHS = 6 nm

RHS = 14 nm


Consider case of hard spheres w/ no charge (3 dk s)

Effect of polydispersity

= 0

= 0.6


Simulated data for homogeneous spheres ( = 0.15, RHS = 10 nm, = 0.4)


Simulated data for homogeneous 11 nm x 21 nm cylinders( = 0.15, RHS = 12 nm, = 0.4)


Simulated data for non-homogeneous spheres ( = 0.285, RHS = 10 nm, = 0.3)


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

Generalized Indirect Fourier Transformation (GIFT)

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