1. Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering of interacting particles. II. Generalized indirect Fourier transformation under consideration of the effective structure factor for polydisperse systems ) Previous GIFT actually assumed a simplistic model for structure factor – the averaged structure factor

2. Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering of interacting particles. II. Generalized indirect Fourier transformation under consideration of the effective structure factor for polydisperse systems ) Previous GIFT actually assumed a simplistic model for structure factor – the averaged structure factor for monodisperse particles Now consider another model - the "effective structure factor" for hard spheres with a better treatment of polydispersity

3. Generalized Indirect Fourier Transformation (GIFT) For monodisperse, homogeneous, isotropic dispersion of spherical particles

4. Generalized Indirect Fourier Transformation (GIFT) For monodisperse, homogeneous, isotropic dispersion of spherical particles Suppose mixture of m components - the components here are different-sized homogeneous spheres Each sphere has a unique form amplitude ƒ  at q = 0 normalized form amplitude B  so that (Blum & Stell, 1979)

5. Generalized Indirect Fourier Transformation (GIFT) For monodisperse, homogeneous, isotropic dispersion of spherical particles Suppose mixture of m components - the components here are different-sized homogeneous spheres Each sphere has a unique form amplitude ƒ  at q = 0 normalized form amplitude B  For this system (Blum & Stell, 1979) structure factor now for inter- action of different-sized spheres

6. Generalized Indirect Fourier Transformation (GIFT) For monodisperse, homogeneous, isotropic dispersion of spherical particles Suppose mixture of m components Then define an averaged form factor x  = molar fraction of  so that

7. Generalized Indirect Fourier Transformation (GIFT) For monodisperse, homogeneous, isotropic dispersion of spherical particles Suppose mixture of m components Then define an averaged form factor x  = molar fraction of  so that

8. Generalized Indirect Fourier Transformation (GIFT) Suppose mixture of m components Then define an averaged form factor so that Thus Note that S eff (q) depends on both the particle interactions & the particle form amplitudes

9. Generalized Indirect Fourier Transformation (GIFT) Note that S eff (q) depends on both the particle interactions & the particle form amplitudes Previously, averaged structure factor used for S eff (q) (weighted addition of partial structure factors S  (q) for a monodisperse system of particles , each having a different radius)

10. Generalized Indirect Fourier Transformation (GIFT) Other models a. local monodisperse approximation accounts for dependence on f, B, but not correlations betwn different-sized particles

11. Generalized Indirect Fourier Transformation (GIFT) Other models a. local monodisperse approximation b. decoupling approximation R(q) accounts for the different scattering properties of the particles Monodisperse S(q) corrected by 'incoherent scattering' term R(q)

12. Generalized Indirect Fourier Transformation (GIFT) Other models a. local monodisperse approximation b. decoupling approximation To calculate S  (q), use mean spherical approx n (Percus & Yevick,1958)

13. Generalized Indirect Fourier Transformation (GIFT) Simulation tests: simulate P(q), S(q) smear add noise get I(q)

14. Generalized Indirect Fourier Transformation (GIFT) Simulation tests: simulate P(q), S(q) smear add noise get I(q) determine initial values for d k s for S(q) then get c s from

15. Generalized Indirect Fourier Transformation (GIFT) Simulation tests: simulate P(q), S(q) smear add noise get I(q) determine initial values for d k s for S(q) then get c s from determine d k s from above iterate until final c s and d k s obtained

16. Generalized Indirect Fourier Transformation (GIFT) Tests 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

17. Generalized Indirect Fourier Transformation (GIFT) Compare S eff (q) for polydispersed system of homogeneous spheres w/  = 0.3,  = 0.3 S lma P-Y S eff

18. Generalized Indirect Fourier Transformation (GIFT) Compare S eff (q) & S ave (q) for polydispersed system of homogeneous spheres – form factor assumed for homogeneous sphere w/ R = 10 nm

19. Generalized Indirect Fourier Transformation (GIFT) Core/shell system

20. Generalized Indirect Fourier Transformation (GIFT) Core/shell system note strong dependence of S eff (q) on polydispersity at low q

21. Generalized Indirect Fourier Transformation (GIFT) Core/shell system S lma P-Y S eff