generalized indirect fourier transformation (gift) (see b. weyerich, j. brunner-popela & o....
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![Page 1: Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering](https://reader035.vdocuments.site/reader035/viewer/2022062804/56649eb55503460f94bbe39f/html5/thumbnails/1.jpg)
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
![Page 2: Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering](https://reader035.vdocuments.site/reader035/viewer/2022062804/56649eb55503460f94bbe39f/html5/thumbnails/2.jpg)
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
![Page 3: Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering](https://reader035.vdocuments.site/reader035/viewer/2022062804/56649eb55503460f94bbe39f/html5/thumbnails/3.jpg)
Generalized Indirect Fourier Transformation (GIFT)
For monodisperse, homogeneous, isotropic dispersion of spherical particles
![Page 4: Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering](https://reader035.vdocuments.site/reader035/viewer/2022062804/56649eb55503460f94bbe39f/html5/thumbnails/4.jpg)
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)
![Page 5: Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering](https://reader035.vdocuments.site/reader035/viewer/2022062804/56649eb55503460f94bbe39f/html5/thumbnails/5.jpg)
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
![Page 6: Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering](https://reader035.vdocuments.site/reader035/viewer/2022062804/56649eb55503460f94bbe39f/html5/thumbnails/6.jpg)
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
![Page 7: Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering](https://reader035.vdocuments.site/reader035/viewer/2022062804/56649eb55503460f94bbe39f/html5/thumbnails/7.jpg)
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
![Page 8: Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering](https://reader035.vdocuments.site/reader035/viewer/2022062804/56649eb55503460f94bbe39f/html5/thumbnails/8.jpg)
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
![Page 9: Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering](https://reader035.vdocuments.site/reader035/viewer/2022062804/56649eb55503460f94bbe39f/html5/thumbnails/9.jpg)
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)
![Page 10: Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering](https://reader035.vdocuments.site/reader035/viewer/2022062804/56649eb55503460f94bbe39f/html5/thumbnails/10.jpg)
Generalized Indirect Fourier Transformation (GIFT)
Other models
a. local monodisperse approximation
accounts for dependence on f, B, but not correlations betwndifferent-sized particles
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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)
![Page 12: Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering](https://reader035.vdocuments.site/reader035/viewer/2022062804/56649eb55503460f94bbe39f/html5/thumbnails/12.jpg)
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)
![Page 13: Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering](https://reader035.vdocuments.site/reader035/viewer/2022062804/56649eb55503460f94bbe39f/html5/thumbnails/13.jpg)
Generalized Indirect Fourier Transformation (GIFT)
Simulation tests:
simulate P(q), S(q)smearadd noiseget I(q)
![Page 14: Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering](https://reader035.vdocuments.site/reader035/viewer/2022062804/56649eb55503460f94bbe39f/html5/thumbnails/14.jpg)
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
![Page 15: Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering](https://reader035.vdocuments.site/reader035/viewer/2022062804/56649eb55503460f94bbe39f/html5/thumbnails/15.jpg)
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
![Page 16: Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering](https://reader035.vdocuments.site/reader035/viewer/2022062804/56649eb55503460f94bbe39f/html5/thumbnails/16.jpg)
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
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Generalized Indirect Fourier Transformation (GIFT)
Compare Seff(q) for polydispersed system of homogeneous spheres w/ = 0.3, = 0.3
SlmaP-Y Seff
![Page 18: Generalized Indirect Fourier Transformation (GIFT) (see B. Weyerich, J. Brunner-Popela & O. Glatter, J. Appl. Cryst. (1999) 32, 197-209. Small- angle scattering](https://reader035.vdocuments.site/reader035/viewer/2022062804/56649eb55503460f94bbe39f/html5/thumbnails/18.jpg)
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
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Generalized Indirect Fourier Transformation (GIFT)
Core/shell system
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Generalized Indirect Fourier Transformation (GIFT)
Core/shell system
note strong dependenceof Seff(q) on polydispersityat low q
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Generalized Indirect Fourier Transformation (GIFT)
Core/shell system
Slma
P-Y Seff