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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

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Generalized Indirect Fourier Transformation (GIFT)

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  1. 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

  2. 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

  3. 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

  4. 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

  5. Generalized Indirect Fourier Transformation (GIFT) Non-dilute systems

  6. Generalized Indirect Fourier Transformation (GIFT) Non-dilute systems Vectord contains the coefficients dk (k = 1-4) determining the structure factor for the particles volume fraction size (radius) polydispersity parameter particle charge

  7. Generalized Indirect Fourier Transformation (GIFT) Non-dilute systems Then

  8. Generalized Indirect Fourier Transformation (GIFT) Non-dilute systems Then Accounting for smearing

  9. Generalized Indirect Fourier Transformation (GIFT) Non-dilute systems Determine c and dk by usual weighted least squares procedure

  10. 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

  11. 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

  12. Generalized Indirect Fourier Transformation (GIFT) Non-dilute systems Simulation tests: simulate P(q), S(q,d) smear add noise get I(q)

  13. 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

  14. 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

  15. 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 finallyuse c s to get pddf pA(r) dk s directly give info on vol. fract., polydispersity distrib., hard sphere radius, charge

  16. 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

  17. 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

  18. 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

  19. Generalized Indirect Fourier Transformation (GIFT) Non-dilute systems Simulated data for homogeneous spheres ( = 0.15, RHS = 10 nm,  = 0.4)

  20. 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)

  21. Generalized Indirect Fourier Transformation (GIFT) Non-dilute systems Simulated data for non-homogeneous spheres ( = 0.285, RHS = 10 nm,  = 0.3)

  22. Generalized Indirect Fourier Transformation (GIFT) Non-dilute systems Simulated data for non-homogeneous spheres ( = 0.285, RHS = 10 nm,  = 0.3)

  23. Generalized Indirect Fourier Transformation (GIFT) Non-dilute systems Simulated data for non-homogeneous spheres ( = 0.285, RHS = 10 nm,  = 0.3)

  24. 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)

  25. 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

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