Indirect Fourier Transformation (IFT)

Indirect Fourier Transformation (IFT)

Here: h = q(r) = correlation fcnp(r) = distance distribution fcn


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


IFT (see Glatter, Acta Phys. Austr. 47, 83 (1977)): Assume scattering particle has finite dimensions (Rmax)


for r > Rmax , (r) = 0

Distance correlation fcn:

where

I(h) = T1 p(r); T1 is above Fourier transformation

Assume scattering particle has finite dimensions (Rmax)


for r > Rmax , (r) = 0 = p(r)

expand p(r)

where (r) are defined only in interval 0 ≤ r ≤ Rmax



For(r), use so-called "cubic B-splines". B = (r)


Then transform

Similar to


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)


Summary of procedure:

a. estimate Rmax

b. compute (r)

c. Fourier transform (r) ––> (h)

d. calculate I(h), etc.


Examples

model I model III


Examples

Results of calcs


Examples


Examples - results of calcs


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)



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



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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Indirect Fourier Transformation (IFT)