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, 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%