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

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

Generalized Indirect Fourier Transformation (GIFT)

For monodisperse, homogeneous, isotropic dispersion of spherical particles

Generalized Indirect Fourier Transformation (GIFT)

For monodisperse, homogeneous, isotropic dispersion of spherical particles

Suppose mixture of m components - the components hereare different-sized homogeneous spheres

Each sphere has a uniqueform amplitude ƒ at q = 0

normalized form amplitude B

so that

(Blum & Stell, 1979)

Generalized Indirect Fourier Transformation (GIFT)

For monodisperse, homogeneous, isotropic dispersion of spherical particles

Suppose mixture of m components - the components hereare different-sized homogeneous spheres

Each sphere has a uniqueform 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

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

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

Generalized Indirect Fourier Transformation (GIFT)

Suppose mixture of m components

Then define an averaged form factor

so that

Thus

Note that Seff(q) depends on both the particle interactions &the particle form amplitudes

Generalized Indirect Fourier Transformation (GIFT)

Note that Seff(q) depends on both the particle interactions &the particle form amplitudes

Previously, averaged structure factor used for Seff(q)

(weighted addition of partial structure factors S(q) for a monodisperse

system of particles , each having a different radius)

Generalized Indirect Fourier Transformation (GIFT)

Other models

a. local monodisperse approximation

accounts for dependence on f, B, but not correlations betwndifferent-sized particles

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)

Generalized Indirect Fourier Transformation (GIFT)

Other models

a. local monodisperse approximation

b. decoupling approximation

To calculate S(q), use mean spherical approxn (Percus & Yevick,1958)

Generalized Indirect Fourier Transformation (GIFT)

Simulation tests:

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

Generalized Indirect Fourier Transformation (GIFT)

Simulation tests:

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

determine initial values for dk s for S(q)then get c s from

Generalized Indirect Fourier Transformation (GIFT)

Simulation tests:

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

determine initial values for dk s for S(q)then get c s from

determine dk s from above

iterate until final c s and dk s obtained

Generalized Indirect Fourier Transformation (GIFT)

Tests

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)

Compare Seff(q) for polydispersed system of homogeneous spheres w/ = 0.3, = 0.3

SlmaP-Y Seff

Generalized Indirect Fourier Transformation (GIFT)

Compare Seff(q) & Save (q) for polydispersed system of homogeneous spheres

– form factor assumed for homogeneous sphere w/ R = 10 nm

Generalized Indirect Fourier Transformation (GIFT)

Core/shell system

Generalized Indirect Fourier Transformation (GIFT)

Core/shell system

note strong dependenceof Seff(q) on polydispersityat low q

Generalized Indirect Fourier Transformation (GIFT)

Core/shell system

Slma

P-Y Seff