Indirect Fourier transformation

In small-angle scattering on single molecules, an intensity $I(\backslash mathbf\{r\})$ is measured and is a function of the magnitude of the scattering vector $q\; =\; |\backslash mathbf\{q\}|\; =\; 4\backslash pi\; \backslash sin(\backslash theta)/\backslash lambda$, where $2\backslash theta$ is the scattered angle, and $\backslash lambda$ is the wavelength of the incoming and scattered beam (elastic scattering). $q$ has units 1/length. $I(q)$ is related to the so-called pair distance distribution $p(r)$ via Fourier Transformation. $p(r)$ is a (scattering weighted) histogram of distances $r$ between pairs of atoms in the molecule. In one dimensions ($r$ and $q$ are scalars), $I(q)$ and $p(r)$ are related by:

:$I(q)\; =\; 4\backslash pi\; n\backslash int\_\{-\backslash infty\}^\backslash infty\; p(r)e^\{-iqr\backslash cos(\backslash phi)\}dr$

:$p(r)\; =\; \backslash frac\{1\}\{2\backslash pi^2n\}\backslash int\_\{-\backslash infty\}^\backslash infty\backslash hat\; (qr)^2\; I(q)e^\{-iqr\backslash cos(\backslash phi)\}dq$

where $\backslash phi$ is the angle between $\backslash mathbf\{q\}$ and $\backslash mathbf\{r\}$, and $n$ is the number density of molecules in the measured sample. The sample is orientational averaged (denoted by $\backslash langle\; ..\; \backslash rangle$), and the Debye equation {{cite journal | first1 = P. | last1 = Scardi | first2 = S. J. L. | last2 = Billinge | first3 = R. | last3 = Neder | first4 = A. | last4 = Cervellino | title = Celebrating 100 years of the Debye scattering equation | journal=Acta Crystallogr A | year=2016| volume=72 | issue = 6 | pages=589–590 | doi=10.1107/S2053273316015680| pmid = 27809198 | doi-access=free | hdl = 11572/171102 | hdl-access = free }} can thus be exploited to simplify the relations by

:$\backslash langle\; e^\{-iqr\backslash cos(\backslash phi)\}\backslash rangle\; =\; \backslash langle\; e^\{iqr\backslash cos(\backslash phi)\}\backslash rangle\; =\; \backslash frac\{\backslash sin(qr)\}\{qr\}$

In 1977 Glatter proposed an IFT method to obtain $p(r)$ form $I(q)$,{{cite journal |author=O. Glatter |title=A new method for the evaluation of small-angle scattering data |journal=Journal of Applied Crystallography |year=1977 |volume=10 |issue=5 |pages=415–421 |doi=10.1107/s0021889877013879|bibcode=1977JApCr..10..415G }} and three years later, Moore introduced an alternative method.{{cite journal|year=1980|title=Small-angle scattering. Information content and error analysis|journal=Journal of Applied Crystallography|volume=13|issue=2|pages=168–175 | doi=10.1107/s002188988001179x | author=P.B. Moore|bibcode=1980JApCr..13..168M }} Others have later introduced alternative methods for IFT,{{cite journal |author=S. Hansen, J.S. Pedersen |title = A Comparison of Three Different Methods for Analysing Small-Angle Scattering Data |journal=Journal of Applied Crystallography |year=1991 |volume=24 |issue = 5 |pages=541–548 |doi=10.1107/s0021889890013322|doi-access=free |bibcode = 1991JApCr..24..541H }} and automatised the process {{cite journal |author=B. Vestergaard and S. Hansen| title=Application of Bayesian analysis to indirect Fourier transformation in small-angle scattering|journal=Journal of Applied Crystallography|year=2006| volume=39| issue=6|pages=797–804|doi=10.1107/S0021889806035291}}{{cite journal|year=2012|title=New developments in the ATSAS program package for small-angle scattering data analysis|journal=Journal of Applied Crystallography | volume=45 |issue=2| pages=342–350 | doi=10.1107/S0021889812007662 | author=Petoukhov M. V. and Franke D. and Shkumatov A. V. and Tria G. and Kikhney A. G. and Gajda M. and Gorba C. and Mertens H. D. T. and Konarev P. V. and Svergun D. I.| pmc=4233345 | pmid=25484842|bibcode=2012JApCr..45..342P }}

This is a brief outline of the method introduced by Otto Glatter. For simplicity, we use $n=1$ in the following.

In indirect Fourier transformation, a guess on the largest distance in the particle $D\_\{max\}$ is given, and an initial distance distribution function $p\_i(r)$ is expressed as a sum of $N$ cubic spline functions $\backslash phi\_i(r)$ evenly distributed on the interval (0,$p\_i(r)$):

{{NumBlk|:|

:$p\_i(r)\; =\; \backslash sum\_\{i=1\}^N\; c\_i\backslash phi\_i(r),$

|{{EquationRef|1}}}}

where $c\_i$ are scalar coefficients. The relation between the scattering intensity $I(q)$ and the $p(r)$ is:

{{NumBlk|:|

:$I(q)\; =\; 4\backslash pi\backslash int\_0^\backslash infty\; p(r)\backslash frac\{\backslash sin(qr)\}\{qr\}\backslash text\{d\}r.$

|{{EquationRef|2}}}}

Inserting the expression for p_i(r) (1) into (2) and using that the transformation from $p(r)$ to $I(q)$ is linear gives:

:$I(q)\; =\; 4\backslash pi\backslash sum\_\{i=1\}^N\; c\_i\backslash psi\_i(q),$

where $\backslash psi\_i(q)$ is given as:

:$\backslash psi\_i(q)=\backslash int\_0^\backslash infty\backslash phi\_i(r)\backslash frac\{\backslash sin(qr)\}\{qr\}\backslash text\{d\}r.$

The $c\_i$'s are unchanged under the linear Fourier transformation and can be fitted to data, thereby obtaining the coefficients $c\_i^\{fit\}$. Inserting these new coefficients into the expression for $p\_i(r)$ gives a final $p\_f(r)$. The coefficients $c\_i^\{fit\}$ are chosen to minimise the $\backslash chi^2$ of the fit, given by:

:$\backslash chi^2\; =\; \backslash sum\_\{k=1\}^\{M\}\backslash frac\{[I\_\{experiment\}(q\_k)-I\_\{fit\}(q\_k)]^2\}\{\backslash sigma^2(q\_k)\}$

where $M$ is the number of datapoints and $\backslash sigma\_k$ is the standard deviations on data point $k$. The fitting problem is ill posed and a very oscillating function would give the lowest $\backslash chi^2$ despite being physically unrealistic. Therefore, a smoothness function $S$ is introduced:

:$S\; =\; \backslash sum\_\{i=1\}^\{N-1\}(c\_\{i+1\}-c\_i)^2$.

The larger the oscillations, the higher $S$. Instead of minimizing $\backslash chi^2$, the Lagrangian $L\; =\; \backslash chi^2\; +\; \backslash alpha\; S$ is minimized, where the Lagrange multiplier $\backslash alpha$ is denoted the smoothness parameter.

The method is indirect in the sense that the FT is done in several steps: $p\_i(r)\; \backslash rightarrow\; \backslash text\{fitting\}\; \backslash rightarrow\; p\_f(r)$.

• Frequency spectrum
• Least-squares spectral analysis

{{DEFAULTSORT:Indirect Fourier Transform}}

Category:Fourier analysis