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 Vector d contains the coefficients d k (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 d k by usual weighted least squares procedure

10. Generalized Indirect Fourier Transformation (GIFT) Non-dilute systems Determine c s and d k 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 d k s by usual weighted least squares procedure Complex problem, so separate into 2 parts. Use a fixed d to 1st get c sthen use fixed c s to get d k 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 d k 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 d k s then get c s from determine d k s from above iterate until final c s and d k s obtained

15. Generalized Indirect Fourier Transformation (GIFT) Non-dilute systems determine initial values for d k s then get c s from determine d k s from above iterate until final c s and d k s obtained finally use c s to get pddf p A (r) d k 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 d k 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 d k s) Effect of radius R HS R HS = 6 nm R HS = 14 nm

18. Generalized Indirect Fourier Transformation (GIFT) Non-dilute systems Consider case of hard spheres w/ no charge (3 d k s) Effect of polydispersity   = 0  = 0.6

19. Generalized Indirect Fourier Transformation (GIFT) Non-dilute systems Simulated data for homogeneous spheres (  = 0.15, R HS = 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, R HS = 12 nm,  = 0.4)

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

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

23. Generalized Indirect Fourier Transformation (GIFT) Non-dilute systems Simulated data for non-homogeneous spheres (  = 0.285, R HS = 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, R HS = 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