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Indirect Fourier Transformation (IFT)

Learn to determine particle size distribution from small-angle scattering data through Indirect Fourier Transformation (IFT) method. Understand normalized scattered intensity and size distribution functions for particles. Explore examples with different particle shapes and size distributions.

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Indirect Fourier Transformation (IFT)

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  1. Indirect Fourier Transformation (IFT) • (see Glatter, J. Appl. Cryst. (1980) 13, 7-11. Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method) • Particles of identical shape, different sizes • (hR) = normalized scattered intensity from single particle, size R • Dn(R) = particle size distribution fcn • m(R) = integral of xs scatteringlength density particle • of size R • Infinite dilution, random orientation

  2. Indirect Fourier Transformation (IFT) • (see Glatter, J. Appl. Cryst. (1980) 13, 7-11. Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method) • Particles of identical shape, different sizes • (hR) = normalized scattered intensity from single particle, size R • Dn(R) = particle size distribution fcn • m(R) = integral of xs scattering length density particle • of size R • To get Dn(R) directly, must have known particle shape and • shape factor (hR)

  3. Indirect Fourier Transformation (IFT) • Particles of identical shape, different sizes • (hR) = normalized scattered intensity from single particle, size R • Dn(R) = particle size distribution fcn • m(R) = integral of xs scattering length density particle • of size R • Alternatively, use IFT method

  4. Indirect Fourier Transformation (IFT) • Particles of identical shape, different sizes • (hR) = normalized scattered intensity from single particle, size R • Dn(R) = particle size distribution fcn • m(R) = integral of xs scattering length density particle • of size R • Alternatively, use IFT method

  5. Indirect Fourier Transformation (IFT) • Particles of identical shape, different sizes • (hR) = normalized scattered intensity from single particle, size R • Dn(R) = particle size distribution fcn • m(R) = integral of xs scattering length density particle • of size R • Alternatively, use IFT method • (solve for by cs by least squares)

  6. Indirect Fourier Transformation (IFT) Example - spheres w/ polynomial size distrib starting distrib fcn x distrib fcn calc'd from scatt data

  7. Indirect Fourier Transformation (IFT) Example - spheres w/ polynomial size distrib starting distrib fcn x distrib fcn calc'd from scatt data smeared scatt data

  8. Indirect Fourier Transformation (IFT) Example - spheres w/ double Gaussian size distrib starting distrib fcn distrib fcn calc'd from scatt data smeared scatt data

  9. Indirect Fourier Transformation (IFT) Previous examples used known particle shape factor If wrong shape factor used (say, for prolate ellipsoids w/ axes R and 3R): distrib fcn forspheres distrib fcn for ellipsoids

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