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Indirect Fourier Transformation (IFT)

Indirect Fourier Transformation (IFT). Here: h = q  (r) = correlation fcn p(r) = distance distribution fcn. Indirect Fourier Transformation (IFT). IFT (see Glatter, Acta Phys. Austr. 47, 83 (1977)) : Assume scattering particle has finite dimensions ( R max )

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Indirect Fourier Transformation (IFT)

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  1. Indirect Fourier Transformation (IFT) Here: h = q (r) = correlation fcn p(r) = distance distribution fcn

  2. Indirect Fourier Transformation (IFT) IFT (see Glatter, Acta Phys. Austr. 47, 83 (1977)): Assume scattering particle has finite dimensions (Rmax) definitely valid for dilute systems for r > Rmax , (r) = 0

  3. Indirect Fourier Transformation (IFT) IFT (see Glatter, Acta Phys. Austr. 47, 83 (1977)): Assume scattering particle has finite dimensions (Rmax) definitely valid for dilute systems for r > Rmax , (r) = 0 Distance correlation fcn: where I(h) = T1 p(r); T1 is above Fourier transformation

  4. Indirect Fourier Transformation (IFT) Assume scattering particle has finite dimensions (Rmax) definitely valid for dilute systems for r > Rmax , (r) = 0 = p(r) expand p(r) where (r) are defined only in interval 0 ≤ r ≤Rmax

  5. Indirect Fourier Transformation (IFT) For(r), use so-called "cubic B-splines". B = (r)

  6. Indirect Fourier Transformation (IFT) Then transform Similar to

  7. Indirect Fourier Transformation (IFT) Determine cs from measured data A(hi) by weighted least squares procedure (M data, N coeffs; wtg values 2(hi), = std. dev.) With cs, can calc I(h), p(r), (r), A(h), (r)

  8. Indirect Fourier Transformation (IFT) Summary of procedure: a. estimate Rmax b. compute (r) c. Fourier transform (r) ––> (h) d. calculate I(h), etc.

  9. Indirect Fourier Transformation (IFT) Examples model I model III

  10. Indirect Fourier Transformation (IFT) Examples Results of calcs

  11. Indirect Fourier Transformation (IFT) Examples

  12. Indirect Fourier Transformation (IFT) Examples - results of calcs

  13. Indirect Fourier Transformation (IFT) Example - scattering curve of a chain molecule calculated from: (see Glatter, J. Appl. Cryst. (1977) 10, 415-421. A new method for the evaluation of small-angle scattering data)

  14. Indirect Fourier Transformation (IFT) Example - scattering curve of a chain molecule calculated from: Rg calc'd 2 ways: a. construct Guinier plot from scattering "data" error in Rg = 25% ln I(h) = ln (v)2 - (Rg2/3) h2

  15. Indirect Fourier Transformation (IFT) Example - scattering curve of a chain molecule calculated from: Rg calc'd 2 ways: a. construct Guinier plot from scattering "data" error in Rg = 25% b. from (D = Rmax) error in Rg = 2%

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