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